Pathspace Connections and Categorical

Geometry

Saikat ChatterjeeInstitute of Mathematical Sciences

Department of MathematicsChennai 600 113 Tamil Nadu, India

email:saikat.chat01@gmail.com

Amitabha LahiriS. N. Bose National Centre for Basic SciencesBlock JD, Sector III, Salt Lake, Kolkata 700098

West Bengal, Indiaemail: amitabhalahiri@gmail.com

Ambar N. SenguptaDepartment of MathematicsLouisiana State University

Baton Rouge, Louisiana 70803, USAemail: ambarnsg@gmail.com

October 11, 2013

Abstract

We develop a new differential geometric structure using category the-oretic tools that provides a powerful framework for studying bundles overpath spaces. We study a type of connection forms, given by Chen inte-grals, over path spaces by placing such forms within a category-theoreticframework of principal bundles and connections. A new notion of ‘dec-orated’ principal bundles is introduced, along with parallel transport forsuch bundles, and specific examples in the context of path spaces aredeveloped.

1 Introduction

In this paper we develop a theory of categorical geometry and explore specificexamples involving geometry over spaces of paths. Our first objective is todevelop a framework that encodes special properties, such as parametrization-invariance, of connection forms on path spaces. For this paper we focus on the

1

case of connections over path spaces given by first-order Chen integrals. Wethen develop a theory of ‘decorated’ principal bundles and parallel-transport insuch bundles. These constructions all sit naturally inside a framework of cate-gorical connections that we develop. A background motivation is to develop aframework that provides a unified setting for both ordinary gauge theory, gov-erning interactions between point particles, and higher gauge theory, governingthe interaction of string-like, or higher-dimensional, extended objects.

There is a considerable current literature (as we cite in a paragraph below)combining geometric ideas and category theoretic structures. However, our workoffers several new ideas; these include:

• our formulation of the notion of a categorical connection, closer in spiritto the traditional notion of parallel-transport but more general than theformulations that are in use in the existing literature;

• a new notion of decorated bundles that provides a natural and rich classof examples of categorical bundles.

Our development of a powerful framework of categorical principal bundles isdifferent from other works, with the action of the categorical structure groupplaying a more explicit direct role, analogous to the case of classical principalbundles.

Our constructions are not developed for the sake of abstract constructions,but rather as a natural framework that expresses the essential elements of avariety of examples. These examples include connections on bundles over pathspaces as well as a new but natural notion of ‘decorated’ bundles that we intro-duce. The study of such bundles is motivated by gauge theories in which pointsin a bundle are replaced by paths, on the bundle, decorated by elements of agroup. Our theory of categorical connections then provides a natural frame-work for formulating the notion of parallel-transport of such decorated paths.Decorated principal bundles provide a framework for generating hierarchies ofexamples of categorical bundles and connections over higher dimensional (thatis, iterations of) path spaces.

We now summarize our results through an overview of the paper. All cate-gories we work with are ‘small’: the objects and morphisms form sets.

• In section 2 we study a connection form ω(A,B) on a principal bundle overa path space. This connection form is invariant under reparametrizationof the paths. We then describe the connection form ω(A,B):

ω(A,B)(v) = A(v(t0)

)+ τ

[∫ t1

t0

B (γ′(t), v(t)) dt

], (1.1)

specifying the meaning of all the terms involved here. Briefly and roughlyput, ω(A,B) is a 1-form on path space obtained by a ‘point-evaluation’ ofa traditional 1-form (the first term on the right) and a first-order Chenintegral (the second term on the right)). Next in Proposition 2.2 we prove

2

that this connection form is also invariant under reparametrization ofpaths, and so induces a connection form on the space of reparametrization-equivalence classes of paths. In Proposition 2.2 we show that ω(A,B) doeshave properties analogous to traditional connection forms on bundles.

• In section 3 we consider equivalence classes of paths, identifying pathsthat differ from each other by erasure of backtracked segments (that is, acomposite path c2aac1, where a is the reverse of a, is considered equivalentto c2c1). The results of section 3, especially Theorem 3.1, show that ω(A,B)

specifies a connection form on the space of backtrack-erasure equivalenceclasses of paths. This is the result that links the geometry with categorytheory: it makes it possible to view ω(A,B) as specifying a connection formover a category whose morphisms are backtrack-erased paths. In Theorem3.2 we prove that ω(A,B) respects another common way of identifyingpaths: paths γ1 and γ2 are said to be ‘thin homotopic’ if one can obtain γ2from γ1 by means of a homotopy that ‘wiggles’ γ1 along itself. Thus, ω(A,B)

specifies a connection also over the space of thin-homotopy equivalenceclasses of paths.

• In section 4 we study the notion of a categorical group; briefly, this isa category whose object set and morphism set are both groups. Themain result, Theorem 4.1 establishes the equivalence between categoricalgroups and crossed modules specified by pairs of groups (G,H). Theseresults are known in the literature but we feel it is useful to present thiscoherent account, as there are many different conventions and definitionsused in the literature and our presentation in this section provides us withnotation, conventions, and results for use in later sections. We also includeseveral examples here.

• Section 5 introduces the key notion of a principal categorical bundle P→B, with a categorical group G as ‘structure group ’ and with both P andB being categories. This general framework does not require B to have asmooth structure. We give examples, including one that uses backtrack-erased path spaces. We conclude the section by showing that the notionof ‘reduction’ of a principal bundle carries over to this categorical setting.(In this work we do not explore categorical analogs of local triviality, atopic that is central to most other works in this area.)

• In section 6 we introduce the notion of a decorated principal bundle. Thisgives a useful example of a categorical principal bundle whose structuredepends on the action of a given crossed module. Briefly, we start withan ordinary principal bundle π : P → B equipped with a connectionA; we form a categorical principal bundle whose base category B hasobject set B and backtrack-erased paths on B as morphisms; the bundlecategory P has object set P and morphisms of the form (γ, h), with γbeing any A-horizontal path on P and h running over a group H. Thusthe morphisms are horizontal paths ‘decorated’ with elements of H. The

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result is a categorical principal bundle whose structure categorical groupis specified by the pair of groups G and H, with G acting on objects of Pand a semi-direct product of G and H acting on the morphisms of P.

• In section 7 we introduce the notion of a categorical connection on a cate-gorical principal bundle. We present several examples, and then show, inTheorem 7.1, how to construct a categorical connection on the bundle ofdecorated paths and then, in Theorem 7.2, in a more abstractly decoratedcategorical bundle.

• In section 8 we describe, in a precise way, categories whose morphismsare (equivalence classes of) paths. We do not use the popular practice ofidentifying thin-homotopy equivalent paths and explain how our approachprovides a very convenient framework in which to formulate properties ofparallel-transport such as invariance under reparametrizations, backtracksand thin-homotopies (Proposition 2.2, Theorem 3.1, Theorem 3.2).

• In section 9 we construct categorical connections at a ‘higher’ geometriclevel: here the objects are paths, and the morphisms are paths of paths.

• We present our final and most comprehensive example of a categoricalconnection in section10, where we develop parallel-transport of ‘decorated’paths over a space of paths. Thus the transport of a decorated path (γ, h)is specified through the data (Γ, h, k), where Γ is a path of paths on thebundle, horizontal with respect to a path space connection ω(A,B), h ∈ Hdecorates the initial (or source) path γ0 = s(Γ) for Γ, and k ∈ K encodesthe rule for producing the resulting final decorated path (γ1, h1).

• Section 11 presents a brief account of associated bundles, along withparallel-transport in such bundles, in the categorical framework.

There is a considerable and active literature at the juncture of categorytheory and geometry. Without attempting to review the existing literature wemake some remarks. The relationship between higher categorical structuresand quantum theories was explored extensively by Freed [14]. Works broadlyrelated to ours include those of Baez et al. [3, 4, 5, 6, 7], Bartels [10], Martins andPicken [20, 21, 11], Schreiber and Waldorf [24, 25], Abbaspour and Wagemann[1], Viennot [26], and Wockel [28]. We study neither local trivialization of 2-bundles nor the related notion of gerbes, both of which are explored in the otherworks just cited. Our definition of categorical connection in section 7 appears tobe new. Other notions, such as 2-connections [6] may be found in the literature.Roughly speaking, our approach stays much closer to geometry than categorytheory in comparison to much of the 2-geometry literature; this is because ourprimary objective is to create a framework for the specific type of connectionsgiven by (1.1).

The work of Martins and Picken [20] gives an account of different notionsof categorical connections as well as a precise examination of the role of ‘thin-homotopy equivalence’. They explain how a traditional connection 1-form A

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and a 2-form B (with values in a Lie group H) on a principal G-bundle π :P → M , with suitable equivariance properties, and which are related to eachother through a ‘flatness’ condition give rise to a ‘categorical holonomy’. Ourconstruction of categorical connections is in terms of parallel-transport ratherthan holonomies and we explore several specific and new examples of categoricalconnections, some of which (as explained in the decorated bundle constructionsin section 10) involve connection forms A, A, as well as 1-forms C1 and C2 withvalues in Lie algebras associated to higher crossed modules.

Before proceeding, let us briefly review some language connected with thestandard framework of connections on principal bundles for ease of referencewhen considering the categorical definitions we introduce later.

For a Lie group G, a principal G-bundle π : P → B is a smooth surjection,where P and B are smooth manifolds, along with a free right action of G on P :

P ×G→ P : (p, g) 7→ Rg(p) = pg,

with π(pg) = π(p) for all p ∈ P and g ∈ G, and there is local triviality: everypoint of B has a neighborhood U and there is a diffeomorphism

φ : U ×G→ π−1(U)

such that φ(u, g)h = φ(u, gh) and πφ(u, g) = u for all u ∈ U and g, h ∈ G.In the categorical formulation we will develop in section 5 we will not (in thepresent paper) impose local triviality.

A vector v ∈ TpP is vertical if dπp(v) = 0. A connection ω is a 1-form onP with values in the Lie algebra L(G), and satisfies the following conditions:(i) ω(R′g(p)X) = Ad(g−1)ω(X) for all p ∈ P , X ∈ Tp and g ∈ G, and (ii)

ω(Y (p)) = Y for all p ∈ P and Y ∈ L(G), where Y (p) = ddt |t=0pe

tY . The crucialconsequence of this definition is that ω decomposes each TpP as a direct sumof the vertical subspace ker dπp and the horizontal subspace kerωp in a manner‘consistent’ with the action of G. This leads to the notion of parallel transport:if γ : [t0, t1] → B is a C1 path and u0 is a point on the fiber π−1

(γ(t0)

)then

there is a unique horizontal lift γω : [t0, t1]→ P , that is a C1 path for which

ω (γ′ω(t)) = 0 for all t ∈ [t0, t1], (1.2)

that initiates at u0. It is in terms of the notion of parallel transport that we statethe definition of a categorical connection on a categorical bundle in section 7.The curvature 2-form Fω, containing information on infinitesimal holonomies,is defined to be

Fω = dω +1

2[ω, ω] (1.3)

where, for any L(G)-valued 1-forms η and ζ, the 2-form [η, ζ] is given by

[η, ζ](X,Y ) = [η(X), ζ(Y )]− [η(Y ), ζ(X)]

for all X,Y ∈ TpP and all p ∈ P .

5

2 A connection form over a path space

In this section we prove certain invariance properties of certain connection formsover spaces of paths. Our results are proved for piecewise C1 paths, but for thesake of consistency with results in later sections we frame the discussions (asopposed to the formal statements and proofs) in terms of C∞ paths.

For a manifold X the set of all parametrized paths on X is⋃t0,t1∈R,t0<t1

C∞ ([t0, t1];X) , (2.1)

and we denote by PX the quotient set obtained by identifying paths that differby a constant time-translation reparametrization. Thus, γ : [t0, t1] → X isidentified with γ+a : [t0 − a, t1 − a] → X : t 7→ γ(t + a) in PX. We will oftennot make a notational distinction between γ and its equivalence class [γ] of suchtime-translation reparametrizations, and in using the term ‘parametrized path’we will often not distinguish between time-translation reparametrizations of thesame path.

We work with a connection A on a principal G-bundle

π : P →M,

where G is a Lie group. We denote by

PAP, (2.2)

the set of all A-horizontal C∞ paths γ : [t0, t1] → P , again identifying pathsthat differ by a constant time-translation. (In [12] we used this notation butwithout any identification procedure.) For a given connection A we view PAPin a natural but informal way as a principal G-bundle over PM ; the projectionPAP → PM is the natural one induced by π : P →M , and the right action ofG on PAP is also given simply from the action of G on P .

Consider a path of A-horizontal paths, specified through a C∞map

Γ : [s0, s1]× [t0, t1]→ P : (s, t) 7→ Γ(s, t) = Γs(t),

where s0 < s1 and t0 < t1, such that Γs is A-horizontal for every s ∈ [s0, s1]. In[12, 2.1] it was shown that

A(v(t)) = A(v(t0)) +

∫ t

t0

FA (γ′(s), v(s)) ds for all t ∈ [t0, t1]. (2.3)

where v : [t0, t1]→ TP : t 7→ ∂sΓ(s0, t) is the vector field along γ = Γs0 pointingin the variational direction. We refer to (2.3) as a tangency condition.

Note that (2.3) is meaningful even if the path γ is piecewise C1 and the vectorfield v merely continuous. Condition (2.3) is equivalent to the requirement that

∂A(v(t))

∂t= FA (γ′(t), v(t)) (2.4)

6

hold at every point where γ′(t) exists (which is all t ∈ [t0, t1] except for finitelymany points).

In the following result we show that the tangency condition (2.3) is inde-pendent of parametrizations, in the sense that if v and γ satisfy (2.3) then anyreparametrization of γ along with the corresponding reparametrization of v alsosatisfy (2.3). We denote by

TγPAP. (2.5)

the vector space of all vector fields v along γ satisfying (2.3), with time-translatesbeing identified.

Proposition 2.1 Let A be a connection on a principal G-bundle, and let v be acontinuous vector field along an A-horizontal piecewise C1 path γ : [t0, t1]→ Psatisfying (2.3). Then:

(i) for any strictly increasing piecewise C1 bijective reparametrization func-tion φ : [t′0, t

′1]→ [t0, t1], the vector field v ◦φ, along the path γ ◦φ satisfies

(2.3) with the domain [t0, t1] replaced by [t′0, t′1];

(ii) if π ◦ γ is constant on an open subinterval of [t0, t1] then γ is constant onthat same subinterval, and if the vector field v = π∗v is constant on thatsubinterval then so is v.

Part (ii) stresses how strongly the behavior of the path π ◦ γ controls thebehavior of a vector field v that satisfies the tangency condition (2.3).Proof. (i) Since φ is a strictly increasing bijection [t′0, t

′1] → [t0, t1] we have, in

particular, φ(t′0) = t0. From (2.3) we have:

A (v ◦ φ(t′)) = A (v (t0)) +

∫ φ(t′)

t0

FA (γ′(u), v(u)) du

= A (v(t0)) +

∫ t′

t′0

FA(γ′(φ(s)

), v(φ(s)

))φ′(s)ds (setting u = φ(s))

= A(v(φ(t′0)

))+

∫ t′

t′0

FA ((γ ◦ φ)′(s), v ◦ φ(s)) ds.

(2.6)

In the second line above the change of variables u = φ(s) can be done intervalby interval on each of which φ has a continuous derivative.

(ii) Now suppose γ = π◦γ is constant on an open interval (T, T+δ) ⊂ [t0, t1].Then, for any t ∈ (T, T + δ) the ‘horizontal projection’ π∗γ

′(t) is 0 and the‘vertical part’ A

(γ′(t)

)is also 0 because γ is A-horizontal. Hence γ′(t) = 0 for

all such t and hence γ is constant on this interval. Next suppose v = π∗v isconstant over (T, T + δ). Then from (2.3) we have

A(v(t)

)−A

(v(T )

)=

∫ t

T

FA (γ′(s), v(s)) ds = 0 (2.7)

7

for all t ∈ (T, T + δ) because of the constancy of γ. Thus,

A(v(t)

)= A

(v(T )

)for all t ∈ (T, T + δ). Thus, since the horizontal parts of v(t) and v(T ) arealso assumed to be equal (by constancy of v over (T, T + δ)) it follows that v(t)

equals v(T ) for all t ∈ (T, T + δ). QED

We will study connections on the bundle PAP , and on related bundles, usingcrossed modules: a crossed module (G,H,α, τ) consist of groups G and H, alongwith homomorphisms

τ : H → G and α : G→ Aut(H). (2.8)

satisfying the identities:

τ(α(g)h

)= gτ(h)g−1

α(τ(h)

)(h′) = hh′h−1

(2.9)

for all g ∈ G and h ∈ H.The structure of a crossed module was introduced in 1949 by J. H. C. White-

head [27, sec 2.9], foreshadowed in the dissertation of Peiffer [22] (sometimesmisspelled as ‘Pfeiffer’ in the literature).

When working in the category of Lie groups, we require that τ be smooth,and the map

G×H → H : (g, h) 7→ α(g, h) = α(g)h (2.10)

be smooth. In this case we call (G,H,α, τ) a Lie crossed module.In what follows we work with a Lie crossed module (G,H,α, τ), connection

forms A and A on a principal G-bundle π : P →M . We denote the right actionof G on the bundle space P by

P ×G→ P : (p, g) 7→ pg = Rgp

and the corresponding derivative map by

TpP → TpgP : v 7→ vgdef= dRg|p(v)

for all p ∈ P and g ∈ G. We also work with an L(H)-valued 2-form B on Pwhich is α-equivariant, in the sense that

Bpg(vg, wg) = α(g−1)Bp(v, w) for all g ∈ G, p ∈ P , and v, w ∈ TpP ,(2.11)

and vanishes on vertical vectors in the sense that

B(v, w) = 0 whenever v, w ∈ Tp and π∗v = 0, (2.12)

for any p ∈ P .

8

For a piecewise C1 A-horizontal path γ : [t0, t1] → P and any continuousvector field v along γ we define

ω(A,B)(v) = A(v(t0)

)+ τ

[∫ t1

t0

B (γ′(t), v(t)) dt

]. (2.13)

(In [12] we used v(t1) instead of v(t0) in the definition of ω(A,B), which is

equivalent to changing B to B − FA here.) We view ω(A,B) as a 1-form on aspace of paths on P , expressed as

ω(A,B) = ev∗t0A+ τ(Z) (2.14)

where evt is the evaluation map

evt : PAP → P : γ 7→ γ(t)

and Z is the L(H)-valued 1-form on PAP given by the Chen integral

Z =

∫γ

B. (2.15)

The preceding definitions, but with evt0 replaced by evt1 , were introducedin our earlier work [12].

We note that ω(A,B) is determined by τ(B), rather than by B directly. Later,in section 7, we will consider a richer structure where B itself, rather than τ(B),can play a role. (See for example the remark at the end of section 10 and afterTheorem 7.1.)

The first observation for ω(A,B) is that it remains invariant when paths arereparametrized; thus, it can be viewed as a 1-form defined on the space of equiv-alence classes of paths, with the equivalence relation being reparametrization:

Proposition 2.2 Let (G,H,α, τ) be a Lie crossed module, and A be a connec-tion on a principal G-bundle π : P →M . Let γ : [t0, t1]→ P be a piecewise C1

A-horizontal path in P , and v : [t0, t1]→ P be a continuous vector field along γthat satisfies the constraint (2.3). Let A be a connection form on π : P → M ,B an L(H)-valued α-equivariant 2-form on P vanishing on vertical vectors, andω(A,B) be as in (2.13). Then

ω(A,B)(v) = ω(A,B)(v ◦ φ) (2.16)

for any strictly increasing piecewise C1 bijective function φ : [t′0, t′1] → [t0, t1],

where on the left v is along γ and on the right v ◦ φ is along γ ◦ φ.

The significance of this result is that it implies that ω(A,B) can be viewedas an L(G)-valued 1-form on the bundle space PAP ; in fact, though, for con-venience, we defined PAP using only constant-time translations, ω(A,B) wouldbe defined on a space of paths, with paths that are reparametrizations of eachother by increasing piecewise C1 bijective functions.

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Proof. The proof is by change-of-variables along the lines of the proof of Propo-sition 2.1 (i):

ω(A,B)(v ◦ φ) = A((v ◦ φ)(t′0)

))+ τ

[∫ t′1

t′0

B((γ ◦ φ

)′(u), (v ◦ φ)(u)

)du

]

= A(v(t0)

)+ τ

[∫ φ(t1)

φ(t0)

B(φ′(u)γ′

(φ(u)

), v(φ(u)

))du

]

= A(v(t0)

)+ τ

[∫ φ(t1)

φ(t0)

B(γ′(φ(u)

), v(φ(u)

))φ′(u)du

]

= A(v(t0)

)+ τ

[∫ t1

t0

B (γ′(t), v) dt

](switching to t = φ(u))

= ω(A,B)(v).

(2.17)

QED

Next we check that ω(A,B) has the properties of a connection form on aprincipal G-bundle.

Proposition 2.3 Let (G,H,α, τ) be a Lie crossed module, A and A be con-nection forms on a principal G-bundle π : P → M . Let B be an L(H)-valuedα-equivariant 2-form on P vanishing on vertical vectors, and ω(A,B) be as in(2.13). Then

ω(A,B)(vg) = Ad(g−1)ω(A,B)(v) (2.18)

for all g ∈ G and all continuous vector fields v along any A-horizontal, piecewiseC1, path γ : [t0, t1]→ P . Moreover, if Y is any element of the Lie algebra L(G)and Y is the vector field along γ given by Y (t) = d

du

∣∣u=0

γ(t) exp(uY ), then

ω(A,B)(Y ) = Y. (2.19)

Proof. Both of these statements follow directly from the defining relation

ω(A,B)(v) = A(v(t0)

)+ τ

[∫ t1

t0

B (γ′(t), v(t)) dt

]. (2.20)

If in this equation we use vg instead of v then the first term on the right isconjugated by Ad(g−1) because A is a connection form, whereas in the secondterm the integrand is conjugated by α(g−1) (keeping in mind that vg is vectorfield along γg), and then using the relation τ

(α(g−1)h

)= g−1τ(h)g for all

h ∈ H, from which we have

τ(α(g−1)Z

)= Ad(g−1)Z for all Z ∈ L(H).

This shows that the second term is also conjugated by Ad(g−1). Next, if v = Y ,where Y ∈ L(G) then the integrand in the second term on the right in (2.20)

10

is 0 because B vanishes on vertical vectors and the first term is Y (since A is a

connection form). QED

With notation and framework as above, there is a natural notion of horizontallifts with respect to ω(A,B). Let

Γ : [s0, s1]× [t0, t1]→M : (s, t) 7→ Γs(t)

be a continuous map, viewed as a path of paths in M . By an ω(A,B)-horizontallift of s 7→ Γs we mean a continuous mapping

Γ : [s0, s1]× [t0, t1]→ P : (s, t) 7→ Γs(t),

with π ◦ Γ = Γ, for which each Γs is piecewise C1 and A-horizontal and, fur-thermore, the ‘flow direction vector field’ ∂sΓs(s, ·) is ω(A,B)-horizontal in thesense that:

ω(A,B)

(∂sΓs(s, ·)

)= 0. (2.21)

The connection ω(A,B) has been explored in our previous work [12].The existence and uniqueness of horizontal lifts of C∞ paths Γ relative to

ω(A,B) follow by using standard results on the existence and uniqueness of so-lutions of ordinary differential equations in Lie groups. The key observationneeded here is that to obtain the ω(A,B)-horizontal lift Γ of a given path of

paths Γ : [s0, s1]× [t0, t1]→M : (s, t) 7→ Γs(t), and a given initial C∞ path Γs0 ,one need only specify the motion s 7→ Γs(t0) ∈ P and then each A-horizontalpath t 7→ Γs(t) is completely specified through the initial value Γs(t0) ∈ P .Needless to say the condition C∞ can be relaxed to a lower degree of piecewisesmoothness.

3 Parallel transport and backtrack equivalence

In this section we prove that path space parallel-transport by the connectionω(A,B) induces, in a natural way, parallel-transport over the space of backtrack-erased paths, a notion that we will explain drawing on ideas from the carefultreatment by Levy [18].

We say that a map γ : [t0, t1]→ X into a space X backtracks over [T, T + δ],where t0 ≤ T < T + 2δ < t1, if

γ(T + u) = γ(T + 2δ − u) for all u ∈ [0, δ]. (3.1)

By erasure of the backtrack over [T, T + δ] from γ we will mean the map

[t0, t1 − 2δ] : t 7→

{γ(t) if t ∈ [t0, T ];

γ(t− 2δ) if t ∈ [T + 2δ, t1].(3.2)

Clearly, this is continuous if γ is continuous, and is piecewise C1 if γ is piecewiseC1. (In later sections we will work with C∞ paths that are constant near the

11

initial and final points. When working with the class of such paths we consideronly backtrack erasures that preserve the C∞ nature.)

In the following we identify a parametrized path c1 with a parametrized pathc2 if c2 = c1 ◦ φ, where φ is a strictly increasing piecewise C1 mapping of thedomain of c2 onto the domain of c1. We say that piecewise C1 paths c1 and c2on M are elementary backtrack equivalent if there are piecewise C1 parametrizedpaths a, b, d on M , and a strictly increasing piecewise C1 function φ from someclosed interval onto the domain of c1, and a strictly increasing piecewise C1

function ψ from some closed interval onto the domain of c2, such that

{c1 ◦ φ, c2 ◦ ψ} = {add−1b, ab}, (3.3)

where juxtaposition of paths denotes composition of paths, and for any pathd : [s0, s1]→M the reverse d−1 denotes the path

d−1 : [s0, s1]→M : s 7→ d(s1 − (s− s0)

). (3.4)

Condition (3.3) means that one of the ci is obtained from the other by erasingthe backtracking part dd−1.

We say that piecewise C1 paths b and c are backtrack equivalent, denoted

c 'bt b, (3.5)

if there is a sequence of paths c = c0, c1, . . . , cn = b, where ci is elementarilybacktrack equivalent to ci+1 for i ∈ {0, . . . , n− 1}.

Backtrack equivalence is an equivalence relation, and if two paths are back-track equivalent then so are any of their reparametrizations. Furthermore, if

a 'bt c and b 'bt d

and if the composite ab is defined then so is cd and

ab 'bt cd. (3.6)

By a backtrack-erased path γ we will mean the backtrack equivalence classcontaining the specific path γ.

A tangent vector to a path γ : [t0, t1] → M is normally taken to be avector field v, of suitable degree of smoothness, that has the property thatv(t) ∈ Tγ(t)M , that is, it is a vector field along γ. Since we wish to identifybacktrack equivalent paths we should not allow the vector field v to ‘pry open’the backtracked parts of γ; thus we define a tangent vector v to γ in the space ofbacktrack-equivalence classes of paths, to be a continuous (or suitably smooth)vector field v along γ : [t0, t1]→M , constant near t0 and t1, with the propertythat if γ backtracks over [T, T + δ], then v also has the same backtrack:

v(T + u) = v(T + 2δ − u) for all u ∈ [0, δ]. (3.7)

12

Proposition 3.1 Let A be a connection form on a principal G-bundle π : P →M , where G is a Lie group. Consider an A-horizontal piecewise C1 path γ :[t0, t1]→ P and a continuous vector field v : [t0, t1]→ TP along γ that satisfiesthe tangency condition (2.4). Suppose that the path γ = π ◦ γ backtracks over[T, T + δ] and suppose the vector field v = π∗v along γ also backtracks over[T, T + δ]. Then:

(i) The path γ backtracks over [T, T + δ].

(ii) The vector field v backtracks over [T, T + δ].

(iii) Erasing the backtrack over [T, T+δ] from v produces a vector field along thepath obtained by erasing the backtrack over [T, T+δ] from γ that continuesto satisfy the tangency condition (2.4).

In the following it is useful to keep in mind that a vector w ∈ TpP iscompletely determined by its projection π∗w ∈ Tπ(p)M and its ‘vertical’ part

A(w)∈ L(G) (which can be transferred to an actual vertical vector in TpP by

means of the right action of G on P ).Proof. (i) The backtracking in γ follows because parallel-transport along thereverse of a path is the reverse of the parallel-transport along the path. Hence

γ(T + u) = γ(T + 2δ − u) for all u ∈ [0, δ].

Note that as consequence the velocity on the way back is minus the velocity onthe way out:

γ′(T + u) = −γ′(T + 2δ − u) for all u ∈ [0, δ]. (3.8)

(ii) The vector v(t) is completely determined by its projection v(t) = π∗v(t)and the ‘vertical part’ A(v(t)) ∈ L(G). To prove that v backtracks over [T, T+δ]we need therefore only show that a ◦ v backtracks over [T, T + δ].

Recall the tangency condition (2.4) :

A(v(t)) = A(v(t0)) +

∫ t

t0

FA (γ′(s), v(s)) ds for all t ∈ [t0, t1]. (3.9)

Since FA vanishes on vertical vectors, on the right we can replace v(t) by vh(t),the A-horizontal lift of v(t) as a vector in Tγ(t)P . Hence

A(v(t)

)= A

(v(t0)

)+

∫ t

t0

FA(γ′(s), vh(s)

)ds. (3.10)

Clearly, the horizontal lift vh backtracks to reflect the backtracking (3.7) of v:

vh(T + u) = vh(T + 2δ − u) for all u ∈ [0, δ].

This, together with (3.8), implies that in the integral on the right in (3.10), thecontribution from s = T + u cancels the contribution from s = T + 2δ − u, forall u ∈ [0, δ]. Hence

A(v(T + u)

)= A

(v(T + 2δ − u)

)for all u ∈ [0, δ].

13

(iii) When t ≥ T + 2δ the part of the integral on the right in (3.10) over[T, T + 2δ] is completely erased, and so A

(v(t)

)retains the same value if the

backtracks are erased from γ. From this it follows that the tangency condition(3.10) continues to hold when the backtrack over [T, T + δ] is erased from v and

from γ. QED

We resume working with a crossed module (G,H,α, τ), connections A, A,and an L(H)-valued 2-form B on a principal G-bundle π : P → M , satisfy-ing α-equivariance (2.11) and vanishing on verticals (2.12). The following isa remarkable feature of the connection form ω(A,B), showing that it specifiesparallel-transport over the space of backtrack equivalence classes of paths.

Theorem 3.1 Let (G,H,α, τ) be a Lie crossed module, A and A be connectionson a principal G-bundle π : P → M , and let ω(A,B) be the 1-form given by(2.14). Then ω(A,B) is well-defined as a 1-form on tangent vectors to backtrack

equivalence classes of A-horizontal paths on P in the following sense. Supposeγ : [t0, t1] → M is a piecewise C1 path, and γ0 a path obtained by erasing abacktrack over [T, T + δ] from γ. Let γ be an A-horizontal lift of γ and γ0 theA-horizontal lift of γ0 with the same initial point as γ. Suppose v is a continuousvector field along γ that backtracks over [T, T + δ]. Then

ω(A,B)(v) = ω(A,B)(v0), (3.11)

where v0 is the vector field along γ0 induced by v, through restriction. In par-ticular, ω(A,B)(v) is 0 if and only if ω(A,B)(v0) is also 0.

Proof. Consider an A-horizontal path γ : [t0, t1] → P . Suppose γ = π ◦ γbacktracks over [T, T+δ]. By Proposition 3.1(i), γ also backtracks over [T, T+δ].Hence

γ′(T + u) = −γ′(T + 2δ − u) (3.12)

for all u ∈ [0, δ] for which either side exists. From the expression

ω(A,B)(v) = A (v(t0)) + τ

(∫ t1

t0

B (γ′(t), v(t)) dt

)(3.13)

we see then that in the second term on the right in the integral∫ t1t0· dt we have

a cancellation:∫ T+δ

T

B (γ′(t), v(t)) dt+

∫ T+2δ

T+δ

B (γ′(t), v(t)) dt = 0.

Note also that the endpoint of a path is unaffected by backtrack erasure. Hencethe first term on the right in (3.13) remains unchanged if the backtrack over[T, T + δ] in γ is erased. Thus the value ω(A,B)(v) remains unchanged if thebacktrack of γ over any subinterval of [t0, t1] is erased. In particular, v is ω(A,B)-horizontal if and only if the backtrack-erased version of v is ω(A,B)-horizontal.

QED

14

There is a notion, known as ‘thin homotopy’ equivalence, that is broaderthan backtrack equivalence. A formalization of this notion using ranks of thederivative of the ‘homotopy’ functions was first introduced by Caetano andPicken [11, sec. 4] (who attribute it to Machado; an earlier version was intro-duced by Barrett [8, sec 2.1.1]). Roughly speaking a thin homotopy wrigglesa path back and forth along itself with no ‘transverse’ motion. The followingresult says essentially that parallel-transport by ω(A,B) is trivial along a thinhomotopy.

Theorem 3.2 Let (G,H,α, τ) be a Lie crossed module, A and A connectionforms on a principal G-bundle π : P →M , and let ω(A,B) be as given by (2.14).Consider a C∞ map

Γ : [s0, s1]× [t0, t1]→M : (u, v) 7→ Γu(v)

for which ∂uΓ and ∂vΓ are linearly dependent at each point of [s0, s1]× [t0, t1],and, furthermore, Γ keeps each endpoint of each u-fixed line {u} × [t0, t1] con-stant:

Γ(u, ti) = Γ(s0, ti) for all u ∈ [s0, s1] and i ∈ {0, 1}. (3.14)

Next let Γ0 : [t0, t1] → P be any A-horizontal path that projects to the initialpath Γ0 on M . Then the ω(A,B)-horizontal lift of u 7→ Γu, with Γs0 = Γ0, is thepath

[s0, s1]→ PAP : u 7→ Γu (3.15)

where Γu : [t0, t1]→ P is the A-horizontal lift of Γu with initial point

Γu(t0) = Γ0(t0) (3.16)

for all u ∈ [s0, s1]. Moreover, ∂uΓ and ∂vΓ are linearly dependent at each pointof [s0, s1]× [t0, t1], where Γ is the map [s0, s1]× [t0, t1]→ P : (u, v) 7→ Γu(v).

Proof. From the tangency condition (2.3) we have

A(∂uΓ(u,w)) = A(∂uΓ(u, t0))

+

∫ w

t0

FA(∂vΓ(u, v), ∂uΓ(u, v)

)dv for all w ∈ [t0, t1].

(3.17)

The first term on the right is 0 because Γ(·, t0) is constant by assumption (3.16).We now show that the second term is also zero. By assumption ∂uΓ and ∂vΓare linearly dependent; thus there exist a(u, v), b(u, v) ∈ R, not both zero, suchthat

a(u, v)∂vΓ(u, v) + b(u, v)∂uΓ(u, v) = 0, (3.18)

for all (u, v) ∈ [s0, s1]× [t0, t1]. Observe that

π∗∂vΓ = ∂v(π ◦ Γ) = ∂vΓ

15

and, similarly,π∗∂uΓ = ∂uΓ.

Hencea(u, v)∂vΓ(u, v) + b(u, v)∂uΓ(u, v) ∈ kerπ∗. (3.19)

(We will see shortly that the left side is in fact zero.) Thus, up to addition of

a vertical vector (on which FA vanishes), the vectors ∂vΓ(u, v) and ∂uΓ(u, v)

are linearly dependent and so the 2-form FA is 0 when evaluated on this pairof vectors. Hence

A(∂uΓ(u, v)) = 0, (3.20)

for all (u, v) ∈ [a0, a1] × [t0, t1]. Using this and the A-horizontality of Γu wehave

A(a(u, v)∂vΓ(u, v) + b(u, v)∂uΓ(u, v)

)= 0, (3.21)

which, in combination with (3.19), implies that

a(u, v)∂vΓ(u, v) + b(u, v)∂uΓ(u, v) = 0. (3.22)

This proves that ∂uΓ and ∂vΓ are linearly dependent.From the definition of ω(A,B) in (2.13) we have

ω(A,B)(∂uΓ) = A(∂uΓ(u, t0)

)+ τ

[∫ t1

t0

B(∂vΓ(u, v), ∂uΓ(u, v)

)dv

]. (3.23)

The second term vanishes because of the linear dependence (3.21). The firstterm on the right (3.23) is also 0 because Γu(t0) constant in u ∈ [s0, s1] by

assumption (3.14). Hence the path (3.15) on PAP is ω(A,B)-horizontal. QED

4 2-groups by many names

In this section we organize known notions, results and examples in a mannerthat is useful for our purposes in later sections. These ideas and related topicson categorical groups are discussed in the works [2, 5, 6, 9, 10, 11, 13, 15, 16, 19,20, 23, 27]. There is considerable diversity of terminology and specific definitionsprevalent in the literature. We present a self-contained account of the topic tosave the reader the trouble of searching through the literature to make sure ourproofs are consistent with definitions available elsewhere.

A category K along with a bifunctor

⊗ : K×K→ K (4.1)

forms a categorical group if both Obj(K) and Mor(K) are groups under theoperation ⊗ (on objects and on morphisms, respectively). Sometimes it will bemore convenient to write ab instead of a ⊗ b. Being a functor, ⊗ carries the

16

identity morphism (1a, 1b), where 1x : x→ x denotes the identity arrows on x,to the identity morphism 1ab:

1a ⊗ 1b = 1a⊗b.

and so, taking a to be the identity element e in Obj(K), it follows that 1e is theidentity element in the group Mor(K).

The functoriality of ⊗ implies the ‘exchange law’:

(f ⊗ g) ◦ (f ′ ⊗ g′) = (f ◦ f ′)⊗ (g ◦ g′) (4.2)

for all f, f ′, g, g′ ∈ Mor(K) for which the composites f ◦f ′ and g◦g′ are defined.The following is a curious but useful consequence of the definition of a cat-

egorical group:

Proposition 4.1 Let K be a categorical group, with the group operation writtenas juxtaposition. Then for any morphisms f : a → b and h : b → c in K, thecomposition h ◦ f can be expressed in terms of the product operation in K:

h ◦ f = f1b−1h = h1b−1f. (4.3)

In particular,hk = h ◦ k = kh if t(k) = s(h) = e. (4.4)

Proof. For f : a→ b and h : b→ c we have, on using the exchange law (4.2),

h ◦ f = (1eh) ◦ (f1b−11b) = (1e ◦ f1b−1)(h ◦ 1b) = f1b−1h. (4.5)

Interchanging the order of the multiplication we also have

h ◦ f = (h1e) ◦ (1b1b−1f) = (h ◦ 1b)(1e ◦ 1b−1f) = h1b−1f (4.6)

QED

Here is an alternative formulation of the definition of a categorical group:

Proposition 4.2 If K, with operation ⊗, is a categorical group then the sourceand target maps

s, t : Mor(K)→ Obj(K)

are group homomorphisms, and so is the identity-assigning map

Obj(K)→ Mor(K) : x 7→ 1x.

Conversely, if K is a category for which both Obj(K) and Mor(K) are groups,s, t, and x 7→ 1x are homomorphisms, and the exchange law (4.2) holds then Kis a categorical group.

17

Proof. Suppose K is a categorical group. Consider any morphisms f : a → band g : c → d in Mor(K). Then, by definition of the product category K×K,we have the morphism (f, g) ∈ Mor(K×K) running from the domain (a, c) tothe codomain (b, d):

(f, g) : (a, c)→ (b, d) in Mor(K×K).

Then, since ⊗ is a functor, f ⊗ g runs from domain a⊗ c to target b⊗ d. Thus,

s(f ⊗ g) = s(f)⊗ s(g) and t(f ⊗ g) = t(f)⊗ t(g).

Thus s and t are homomorphisms.Next, for any objects x, y ∈ Obj(K), the identity morphism

(1x, 1y) : (x, y)→ (x, y)

in Mor(K×K) is mapped by the functor ⊗ to the identity morphism

1x⊗y : x⊗ y → x⊗ y,

and this just means that1x ⊗ 1y = 1x⊗y.

Thus x 7→ 1x is a group homomorphism.Conversely, suppose s, t, x 7→ 1x are group homomorphisms and the ex-

change law (4.2) holds (both sides of (4.2) are meaningful because s and t arehomomorphisms). Then (4.2) says that ⊗ maps composites to composites, whilex 7→ 1x being a homomorphism implies that 1x⊗y = 1x⊗ 1y, and so ⊗ is indeed

a functor, preserving composition and mapping identities to identities. QED

See [13, 17] for more on 2-groups, related notions, along with more references(the relation (4.3), proved above, is mentioned in [13].)

The preceding result suggests that when working with groups with morestructure, for example with Lie groups, it would be more natural to continueto require that the source, targets, and identity-assigning maps respect theadditional structure. Thus by a categorical Lie group we mean a category Kalong with a functor ⊗ as above, such that Mor(K) and Obj(K) are Lie groups,and the maps s, t, and x 7→ 1x are smooth homomorphisms.

Example CG1. For any group K we can construct a categorical groupK0 by taking K as the object set and requiring there be a unique morphisma→ b for every a, b ∈ K. At the other extreme we have the discrete categoricalgroup Kd whose object set is K but whose morphisms are only the identitymorphisms. �

Example CG2. Letπ : K → K (4.7)

be a surjective homomorphism of groups. We think of K as a ‘covering group’,or a principal bundle over K, with each fiber π−1(k) standing ‘above’ the pointk. The structure group is

Z = kerπ = π−1(e),

18

with e being the identity element of K. For the category K we take the objectset to be K itself. A morphism a → b is to be thought of as an arrow a → b,with π(a) = a and π(b) = b and such that a→ b is identified with ak → bk, for

every k ∈ kerπ. Thus the discrete categorical group Zd acts on the right on thecategory K0 (notation as in CG1) as follows: an object z ∈ Z acts on the right onan object a ∈ K to produce the product az; a morphism z → z in Mor(Zd) acts

on a morphism a→ b in Mor(K0) to produce the morphism az → bz. (See (5.1)for a precise definition of right action in this context.) We construct a category

K which we view as the quotient K/Zd. The object set of K is K. A morphism

f : a → b for K is obtained as follows: we choose some a ∈ π−1(a), b ∈ π−1(b)

and take f to be the arrow a → b, identifying this with az → bz for all z ∈ Z.More compactly, Mor(K) is the quotient Mor(K)/Mor(Zd) where the action of

the group Mor(Zd) on Mor(K) is given by: (a → b)(z → z) = az → bz. Inorder to make the category K into a categorical group we define a multiplicationon Mor(K) by using the multiplication structure on Mor(K): for a morphism

f : a→ b given by a→ b and g : c→ d given by c→ d we define fg to be

fg = (ac→ bd). (4.8)

For this to be well-defined, the right side should not depend on the choicesx ∈ π−1(x); if we assume that the subgroup Z is central in K, then:

azcw = aczw

holds for all a, c ∈ K and z, w ∈ Z, and so the right side in (4.8) is determinedentirely by f and g. �

A more specific class of examples of categorical groups is provided by takingK to be any compact Lie group and K its universal covering group.

A group K gives rise to a category G(K) in a natural way: G(K) has justone object O, the morphisms of G(K) are the elements of K, all having Oas both source and target, with composition of morphisms being given by thegroup operation in K.

Next we discuss a concrete form of a categorical group. For this recall thenotion of a crossed module (G,H,α, τ) from (2.9).

Theorem 4.1 Suppose G is a categorical group, with the group operation writ-ten as juxtaposition: a⊗ b = ab, and with s, t : Mor(G)→ Obj(G) denoting thesource and target maps. Let H = ker s. Let τ : H → G and α : G→ Aut(H) bethe maps specified by

τ(θ) = t(θ),

α(g)θ = 1gθ1g−1

(4.9)

for all g ∈ G and θ ∈ H. Then (G,H,α, τ) is a crossed module.Conversely, suppose (G,H,α, τ) is a crossed module. Then there is a cate-

gory G whose object set is G and for which a morphism h : a→ b is an orderedpair

(h, a),

19

where h ∈ H satisfiesτ(h)a = b,

with composition being given by

(h2, b) ◦ (h1, a) = (h2h1, a). (4.10)

Moreover, G is a categorical group, with group operation on Obj(G) being theone on G, and the group operation on Mor(G) being

(h, a)(k, c) = (hα(a)(k), ac) . (4.11)

Stated in technical terms, we have an equivalence between the category ofcrossed modules and the category of categorical groups, defined in the obviousway.

Let (G,H,α, τ) be the crossed module constructed as above from a cate-gorical group G, as above. Let K be the categorical group constructed from(G,H,α, τ) according to the prescription in Theorem 4.1. The objects of K arejust the elements of G. A morphism in Mor(K) is a pair(

s(φ), φ1s(φ)−1

), (4.12)

where φ ∈ Mor(G), so that

φ1s(φ)−1 ∈ H = ker s. (4.13)

Proof. Assuming that G is a categorical group we will show that (G,H,α, τ)is a crossed module. Let τ be the target map t restricted to the subgroup H ofMor(G)

τ : H → G : h 7→ τ(h) = t(h). (4.14)

Next for g ∈ G let α(g) : H → H be given on any h ∈ H by

α(g)h = 1gh1g−1 . (4.15)

Note that the source of α(g)h is

s(α(g)h

)= geg−1 = e,

where e is the identity element of Obj(G). Thus α(g)h is indeed an element ofH. Moreover, it is readily checked that α(g) is an automorphism of H.

The target of α(g)h is

τ(α(g)h

)= t(α(g)h

)= gτ(h)g−1. (4.16)

Next, consider h, h′ ∈ H, and suppose τ(h) = g and τ(h′) = g′. Then

α(τ(h)

)(h′) = 1gh

′1g−1 = hh′h−1, (4.17)

the last equality of which can be verified by using Proposition 4.1 (specifically(4.4)), which implies that the element h−11g : g 7→ e commutes with h′ : e →

20

g′. Equations (4.16) and (4.17) are exactly the conditions (2.9) that make(G,H,α, τ) a crossed module.

Before proceeding to the proof of the converse part, we observe that a mor-phism f ∈ Mor(G) is completely specified by its source a = s(f) and by themorphism

h = f1a−1 ∈ ker s.

Now suppose (G,H,α, τ) is a crossed module. We construct a category Kwith object set G. If f : a → b is to be a morphism then f1a−1 would be amorphism e → ba−1. Thus, for a, b ∈ G we take a morphism a → b to bespecified by the source a along with an element h ∈ H for which τ(h) = ba−1.So we define a morphism f ∈ Mor(K) to be an ordered pair (h, a) ∈ H × G,with source and target given by

s(f) = a and t(f) = τ(h)a. (4.18)

Thus the source and target maps are

s(h, a) = a and t(h, a) = τ(h)a. (4.19)

To understand what the composition law should be, recall from (4.3) that forany morphisms f : a→ b and g : b→ c we have

(g ◦ f)1a−1 = g1b−1f1a−1 . (4.20)

Thus we can define the composition of morphisms (h1, a), (h2, b) ∈ H ×G, withb = τ(h1)a, by

(h2, b) ◦ (h1, a) = (h2h1, a). (4.21)

The identity morphism 1a is then (e, a) because

(h2, a) ◦ (e, a) = (h2, a) and (e, c) ◦ (h1, b) = (h1, b),

where c = τ(h1)b. Associativity of composition is clearly valid. Thus K isindeed a category. It remains to define a product on Mor(K) and prove thatthis product is functorial.

For f : a→ b and g : c→ d we have

(fg)1(ac)−1 = f1a−11ag1c−11a−1

which motivates us to define the product on Mor(K) by

(h, a)(k, c) = (hα(a)(k), ac) . (4.22)

If we identify H and G with subsets of Mor(K) through the injections

H → Mor(K) : h 7→ (h, e)

G→ Mor(K) : a 7→ (e, a)(4.23)

21

then, using (4.22), the product ha corresponds to the element

ha = (h, e)(e, a) = (h, a). (4.24)

Hence the multiplication operation (4.22) takes the form

hakc = hα(a)(k) ac, (4.25)

which means the commutation relation

ah = α(a)(h) a for all a ∈ G and h ∈ H, (4.26)

or, equivalently:

α(a)(h) = aha−1 for all a ∈ G and h ∈ H. (4.27)

The product (4.22) is a standard semi-direct product of groups, and it is astraightforward, if lengthy, calculation to verify that Mor(K) is indeed a groupunder the operation:

Mor(K) = H oα G. (4.28)

It is readily checked that the source and target maps s and t given by (4.19) arehomomorphisms, and so is the map

Obj(K)→ Mor(K) : a 7→ 1a = (e, a). (4.29)

Thus K is a categorical group.It is straightforward to verify the exchange law (4.2) holds. Finally, by

construction, G = Obj(K), and H is identifiable as the subgroup of Mor(K)

given by ker s. QED

Example CG3. Let Lo be the set of all backtrack erased piecewise C1

loops based at a point o in a manifold B; under composition, this is clearly agroup. Now the method of example CG2 can be used, with K = Lo, to formcategorical groups with object group Lo. �

By a 2-category C2 over a category C1 we mean a category C2 whose objectsare the arrows of C1:

Obj(C2) = Mor(C1).

Let K be a categorical group, and K the group formed by Obj(K). Then K isa 2-category over G(K), which is the categorical group whose object group hasjust one element and whose morphism group is K. (For an extensive account ofthe theory of 2-categories see Kelly and Street [17].)

Let G1 and G2 be categorical groups such that Obj(G2) = Mor(G1). Let(G,H,α1, τ1) be the crossed module associated with G1, and (Hoα1G,K,α2, τ2)the crossed module associated with G2. We identify H and G naturally withsubgroups of H oα1

G, so that each element of this semi-direct product can beexpressed as hg, with h ∈ H and g ∈ G. The following computation will beuseful later.

22

Lemma 4.2 With notation as above,

α2

(α1(g−11 )(h−11 h)

)(α2(g−11 )(k)

)= α2(g−11 h−11 h)(k), (4.30)

for all g1 ∈ G1, h1, h ∈ H, and k ∈ K.

Proof. Recall from (4.26) the commutation rule:

gh = α1(g)(h) g for all g ∈ G and h ∈ H. (4.31)

Then we have

α2

(α1(g−11 )(h−11 h)

)(α2(g−11 )(k)

)= α2

(α1(g−11 )(h−11 h)g−11

)(k)

= α2(g−11 h−11 h)(k) (using (4.31) with g = g−11 ),

(4.32)

which proves (4.30). QED

5 Principal categorical bundles

In the traditional topological description, a principal bundle π : P → B is asmooth surjection of manifolds, along with a Lie group G acting freely on P onthe right, the action being transitive on each fiber π−1(b) (we do not considerlocal triviality here). In this section we formulate a categorical description thatincludes a wider family of geometric objects than is included in the traditionalnotion of principal bundles. In particular, the framework of categorical bundlesincludes bundles over path spaces with a pair of groups serving as structuregroups, one at the level of objects and one at the level of morphisms. Categoricalbundles, in different formalizations, have been studied in the literature (forexample in Baez et al. [3, 4, 5, 6, 7], Bartels [10], Schreiber and Waldorf [24, 25],Abbaspour and Wagemann [1], Viennot [26], and Wockel [28]). Our explorationis distinct and our focus is more on the geometric side than on category theoreticexploration. However, unlike many other works in the area (such as the analysisby Wockel [28] of the classification 2-bundles in terms of Cech cohomology), wedo not explore any notions of local triviality nor do we impose any smoothstructure on the base categories in the formal definition.

Let P be a category and Z a categorical group. Denote by P the set ofobjects of P and by Z the set of objects of Z:

P = Obj(P) and Z = Obj(Z).

By a right action of Z on P we mean a functor

P× Z→ P : (x, g) 7→ ρ(g)x = ρ(x, g) (5.1)

23

that is a right action both at the level of objects and at the level of morphisms;thus, both

Obj(P)×Obj(Z)→ Obj(P) : (A, g) 7→ ρ(A, g)

Mor(P)×Mor(Z)→ Mor(P) : (F, φ) 7→ ρ(F, φ)(5.2)

are right actions. We assume, moreover, that both these actions are free.Note that functoriality of ρ implies, in particular, that

s(ρ(F, φ)

)= ρ(s(F ), s(φ)

)t(ρ(F, φ)

)= ρ(t(F ), t(φ)

) (5.3)

By analogy with principal bundles we define a principal categorical bundleto be a functor π : P→ B along with a right action of a categorical group Z onP satisfying the following conditions:

(i) π is surjective both at the level of objects and at the level of morphisms;

(ii) the action of Z on P is free on both objects and morphisms;

(iii) the action of Obj(Z) on the fiber π−1(b) is transitive for each object b ∈Obj(B), and the action of Mor(Z) on the fiber π−1(φ) is transitive foreach morphism φ ∈ Mor(B).

Notice, however, that we are not imposing any form of local triviality.There is a consistency property of compositions in P with respect to the

action of Z.

Lemma 5.1 Suppose ρ is a right action of a categorical group Z on a categoryP, the action being free on both objects and morphisms. Let F and H be mor-phisms in Mor(P), with the composition H ◦ F defined, and let F ′ and H ′ bemorphisms in Mor(P), with F ′ being in the Mor(Z)-orbit of F and H ′ in theMor(Z)-orbit of H, and the composite H ′ ◦F ′ also defined; then the compositionH ′ ◦ F ′ is in the Mor(Z)-orbit of H ◦ F .

Proof. Suppose F ′ is in the Mor(Z)-orbit of F , and H ′ is in the Mor(Z)-orbitof H. Then

F ′ = Fρ(φ) and H ′ = Hρ(ψ) (5.4)

for some φ, ψ ∈ Mor(Z). The composability H ′ ◦ F ′ implies that the target ofFρ(φ) = ρ(F, φ) is the source of Hρ(ψ), and so, by (5.3),

ρ(t(F ), t(φ)

)= t (ρ(F, φ)) = s (ρ(H,ψ)) = ρ

(s(H), s(ψ)

). (5.5)

Now t(F ) = s(H) when H ◦ F is defined; hence by (5.5) and by freeness of theaction ρ on Mor(P) we conclude that

t(φ) = s(ψ),

24

and so the composite ψ ◦ φ is defined. Then, using the functoriality of thecategorical group action ρ, we have

H ′ ◦ F ′ = ρ(H,ψ) ◦ ρ(F, φ) = ρ(H ◦ F,ψ ◦ φ) (5.6)

which shows that H ′ ◦ F ′ is in the Mor(Z)-orbit of H ◦ F . Hence composition

is well-defined on the quotient Mor(B) as specified in (5.7). QED

The consistency property of composition allows us to form a ‘quotient’ cat-egory P/Z.

Theorem 5.2 Suppose ρ is a right action of a categorical group Z on a categoryP, the action being free on both objects and morphisms.

Let B be the category whose object set is B = Obj(P)/Obj(Z), and whosemorphisms are Mor(Z)-orbits of morphisms in Mor(P), with composition de-fined by

[H ◦ F ] = [H] ◦ [F ], (5.7)

where [· · · ] denotes the Mor(Z)-orbit. Then

π : P→ B

taking every object X ∈ Obj(P) to the Obj(Z)-orbit of X, and every morphismF to its Mor(Z)-orbit is a functor, and, along with the right action of Z on P,specifies a principal categorical bundle.

Proof. LetB = Obj(P)/Obj(Z),

andπP : P → B

the quotient map. For x ∈ B the identity morphism 1x is the orbit of 1X , forany X ∈ P whose orbit is x. Taking B as the set of objects, and morphisms asjust described, we have a category B.

It is clear from the construction of morphisms and compositions in B thatthe quotient association

π : P→ B

is a functor. Moreover, again from the construction of B in terms of orbits of theaction on Z, the right action of Z on P is transitive on fibers. By hypothesis,the action is also free. Hence π, along with the action, specifies a principal

categorical bundle. QED

We can revisit Example CG2 in light of the preceding theorem. For thiswe take P to be the category K0 whose objects are the elements of a givengroup K, and whose morphisms are all ordered pairs of elements of K; let Z bea subgroup of K = Obj(K), acting by right multiplication on K, and Zd thecategorical group in which the only morphisms are the identities z → z withz running over Z. Then, as in Example CG2, there is a category K = K/Zd.

25

This provides a categorical principal bundle K→ K with structure categoricalgroup Zd (for this there is no need to assume Z is central).

Example P1. A traditional principal G-bundle π : P → B generates acategorical principal bundle in the following way. Take P to be the discretecategory with object set P (in a discrete category the only morphisms are theidentity morphisms), B to be the discrete category with object set B, and Gd

to be the categorical group whose object set is G and whose only morphismsare the identity morphisms. Then we have a principal categorical bundle.

Example P2. A more interesting example is obtained again from a principalG-bundle π : P → B, but with the categorical group (Example CG1) being G0,whose object set is G and for which there is a unique morphism g1 → g2 forevery g1, g2 ∈ G; we denote this morphism by (g1, g2). Take B to be the categorywith object set B and with morphisms being backtrack-erased paths on B. ForP take the category whose object set is P and for which a morphism p → q isa triple

(p, q; γ),

where γ is any backtrack-erased path on B from π(p) to π(q). Define composi-tion of morphisms in the obvious way, and define a right action of G on P bytaking it to be the usual right action of G on P at the level of objects and bysetting

(p, q; γ)(g1, g2) = (pg1, qg2; γ), (5.8)

for all p, q ∈ P , all backtrack-erased paths γ on B from π(p) to π(q) and allmorphisms (g1, g2) in G0. Defining the projection P → B to be π at the levelof objects and (p, q; γ) 7→ γ at the level of morphisms yields then a principalcategorical bundle.

Example P3. Next consider a connection A on a principal G-bundle π :P → B. Let B have object set B, with arrows being backtrack-erased pathsin B. Let Pbt(P )0

Ahave objects the points of P , with arrows being backtrack

erased A-horizontal paths. There is then naturally the projection functor π :P → B. The categorical group Gd whose objects form the group G, andwhose morphisms are all the identity morphisms, has an obvious right actionon Pbt(P )0

A, and thus we have a categorical principal bundle.

In the following section we will explore a more substantive example.First, however, we explore the notion of reduction of a bundle. The basic

example of a principal bundle is that of the frame bundle FrM of a manifold M :a typical point p ∈ FrM is a basis (u1, . . . , un) of a tangent space TmM to Mat some point m, and π : P → M is defined by π(p) = m. The group GL(Rn)acts on the right on FrM by transformations of frames:

(u1, . . . , un) · g = [u1, . . . , un]

g11 g12 . . . g1ng21 g22 . . . g2n...

...... . . .

...gn1 gn2 . . . gnn

.

26

With this structure π : FrM → M is a principal GL(Rn)-bundle. If M isequipped with a metric, say Riemannian, then we can specialize to orthonormalbases; two such bases are related by an orthogonal matrix. Let O(n) be thesubgroup consisting of orthogonal matrices inside GL(Rn). Then we have thebundle OFrM →M of orthonormal frames, and this is a principal O(n)-bundle.There is the natural ‘inclusion’ map OFrM → FrM , which preserves the prin-cipal bundle structures in the obvious way. The general notion here is that of‘reduction’ of a principal bundle. Let π : P → M be a principal G-bundle andβ : Go → G be a homomorphism of Lie groups; then a reduction of π by β toGo is a principal G0-bundle πo : Po →M along with a smooth map

f : Po → P

that maps each fiber π−1o (m) into the fiber π−1(m) and f(pg) = f(p)β(g), forall p ∈ Po and g ∈ Go.

We can define an analogous notion for principal categorical bundles. Supposeπ : P→ B is a principal categorical bundle with structure group G, a categoricalgroup. Next suppose Go is a categorical group and

β : Go → G

is a functor that is a homomorphism on objects as well as morphisms. Thenby a reduction of π : P → B by β we mean a principal categorical bundleπo : Po → B, with structure group Go, along with a functor

f : Po → P

that is fiber preserving both on objects and morphisms and satisfies

f(pg) = f(p)β(g)

both when (p, g) ∈ Obj(P0)×Obj(G0) and when (p, g) ∈ Mor(P0)×Mor(G0).We will see an example of such a reduction in Proposition 6.1 in the next section.

6 A decorated principal categorical bundle

In this section we introduce a new notion, that of a decorating a principalcategorical bundle. This construction will eventually (in section 10, specifi-cally (10.10)) make it possible to start with an ordinary principal G-bundleπ : P → M with connection and a categorical group K, with object group G,and obtain a principal categorical bundle, over a space of paths on M , withstructure categorical group G2 whose object group is Mor(K). We will continueto explore this construction in later sections, in the context of path spaces andin the abstract.

Let K be a categorical group and (G,H,α, τ) the corresponding crossedmodule. Recall that each morphism φ ∈ Mor(K) can be identified with the or-dered pair (h, g) ∈ HoαG where g = s(φ) and h = φ1s(φ)−1 ∈ H, the subgroup

27

of Mor(K) consisting of morphisms that have source the identity element e inG = Obj(K). The composition law for morphisms translates to the product inH, as explained in (4.21).

Let Pbt(M) be the category whose objects are the points of M and whosemorphisms are the backtrack erased piecewise C1 paths on M . (Note that by‘backtrack-erased path’ we mean an equivalence class of paths that are backtrackequivalent to each other.) Instead of working with piecewise C1 paths we couldalso work with C∞ paths that are constant near the initial and final points; thelatter condition ensures that composites of such paths are C∞.

We construct a category Pbt(P )decA

by decorating the morphisms of the cate-gory in Example P3 in section 5 with elements of the group H. Specifically, wedefine the category Pbt(P )A as follows: (i) the object set is P ; (ii) a morphismf is a pair (γ, h), where γ is a piecewise C1 backtrack-erased A-horizontal pathon P , and h ∈ H (corresponding to a morphism φ ∈ Mor(K) with s(φ) = e),source and targets being specified by

s(γ, h) = s(γ) and t(γ, h) = t(γ)τ(h−1). (6.1)

Define composition of morphisms by

(γ2, h2) ◦ (γ1, h1) = (γ3, h2h1), (6.2)

where γ3 is the composite of the path γ1 with the right translate γ2τ(h1) (sothat the final and initial points match correctly):

γ3 = γ2τ(h1) ◦ γ1. (6.3)

Note thatt(γ3, h2h1) = t(γ2)τ(h1)τ(h2h1)−1 = t(γ2, h2), (6.4)

and the corresponding result for the sources is clear.Then Pbt(P )dec

Ais a category, and there is the functor

π : Pbt(P )decA→ Pbt(M) (6.5)

that associates to each object p ∈ P the object π(p) ∈M , and to each morphism(γ, h) associates the backtrack erased path π ◦ γ.

Now we define a right action ρ of the categorical group K on Pbt(P )decA

.At the level of objects this is simply the usual right action of G on P . Formorphisms we define the action as follows. Recall that a morphism in K hasthe form (h1, g1), where

s(h1, g1) = g1, t(h1, g1) = τ(h1)g1.

Then for (γ, h),∈ Mor(Pbt(P )A) and (h1, g1) ∈ Mor(K), we define

ρ ((γ, h), (h1, g1)) = (γ, h) · (g1, h1) =(γg1, α(g−11 )(h−11 h)

), (6.6)

where, on the right, γg1 is the usual right-translate of γ by g1.

28

The following computation shows that (6.6) does define a right action:

[(γ, h) · (h1, g1)] · (h2, g2) =(γg1, α(g−11 )(h−11 h)

)· (h2, g2)

=(γg1g2, α(g−12 )

(h−12 α(g−11 )(h−11 h)

))=(γg1g2, α((g1g2)−1)(α(g1)(h−12 )h−11 h)

)= (γ, h) · (h1α(g1)(h2), g1g2)

= (γ, h) · [(h1, g1) · (h2, g2)] (using (4.22)).

(6.7)

From the definition (6.6) we see directly that (γ, h) ·(h1, g1) is equal to (γ, h)if and only if g1 = e and h1 = e, and so the action is free. It is also readilychecked that the action is transitive on fibers: suppose (γ0, h0) and (γ, h) projectto γ; then γ0 = γg1 for a unique g1 ∈ G, and then on taking

h1 = hα(g1)(h−10 ),

we have(γ0, h0) = (γ, h) · (h1, g1).

For computational purposes we switch back to the regular morphism nota-tion: every morphism in Pbt(P )dec

Ais of the form

(γ, θ) ∈ Mor(Pbt(P )A)× ker s ⊂ Mor(Pbt(P )A)×Mor(K)

where s(θ) = e, the identity in Obj(K). The source and targets are

s(γ, θ) = s(γ) and t(γ, θ) = t(γ)t(θ)−1. (6.8)

We will now check that the expression for the action (6.6) in the morphismnotation is

(γ, θ) · φ = (γs(φ), φ−1θ1s(φ)) (6.9)

where φ−1 : a−1 → b−1 is the multiplicative inverse of φ : a→ b ∈ Mor(K) (werarely need to use the compositional inverse b→ a and avoid introducing a newnotation relying on the context to make the intended meaning clear). To makethe comparison we take

θ = h,

and we recall from (4.12) that a morphism φ of Mor(K) is of the form

(h1, g1) =(φ1s(φ)−1 , s(φ)

). (6.10)

It is clear that the first component on the right side in (6.9) corresponds to thefirst component on the right side in (6.6). For the second component we have

α(g1)−1(h−11 h) = α(s(φ)−1)(

1s(φ)φ−1θ)

= 1s(φ)−1

(1s(φ)φ

−1θ)

1s(φ) (using (4.9))

= φ−1θ1s(φ),

(6.11)

29

which is exactly the first component on the right in (6.9). Thus the right actiongiven by (6.9) is the same as that given by (6.6).

The source and target assignments behave functorially:

s ((γ, θ) · φ) = s(γs(φ), φ−1θ1s(φ),

)= s(γ)s(φ) = s(γ, θ)s(φ)

t ((γ, θ) · φ) = t(γs(φ), φ−1θ1s(φ)

)= t(γ)s(φ)t

(φ−1θ1s(φ)

)−1= t(γ)t(θ)−1t(φ)

= t(γ, θ)t(φ).

(6.12)

Next we check that (6.9) does specify a right action:((γ, θ) · φ1

)· φ2 = (γs(φ1), φ−11 θ1s(φ1)

)· φ2

=(γs(φ1)s(φ2), φ−12 φ−11 θ1s(φ1)1s(φ2)

)= (γ, θ) · (φ1φ2).

(6.13)

(We have already proved this in (6.7) in terms of the crossed module (G,H,α, t).)The composition law (6.2) reads

(γ2, θ2) ◦ (γ1, θ1) = (γ2t(θ1) ◦ γ1, θ2θ1) , (6.14)

whose target is clearly the same as the target of (γ2, θ2) (note that the compositeon the right in (6.14) is defined). We check associativity:

(γ3, θ3) ◦ ((γ2, θ2) ◦ (γ1, θ1)) = (γ3, θ3) ◦ (γ2t(θ1) ◦ γ1, θ2θ1)

= (γ3t(θ2θ1) ◦ γ2t(θ1) ◦ γ1 , θ3θ2θ1) ,(6.15)

while

((γ3, θ3) ◦ (γ2, θ2)) ◦ (γ1, θ1) = (γ3t(θ2) ◦ γ2, θ3θ2) ◦ (γ1, θ1)

=((γ3t(θ2) ◦ γ2

)t(θ1) ◦ γ1 , θ3θ2θ1

),

(6.16)

in agreement with the last expression in (6.15). The existence of identity mor-phisms is readily verified.

We can now check functoriality of the right action by examining

((γ2, θ2) ◦ (γ1, θ1)) · (φ2 ◦ φ1),

wherein note that the composability of φ2 with φ1 means that

s(φ2) = t(φ1). (6.17)

Composing and then acting produces:

((γ2, θ2) ◦ (γ1, θ1)) · (φ2 ◦ φ1) = (γ2t(θ1) ◦ γ1, θ2θ1) · (φ2 ◦ φ1)

=((γ2t(θ1) ◦ γ1

)s(φ1), (φ2 ◦ φ1)−1θ2θ11s(φ1)

),

(6.18)

30

while first acting and then composing produces:

(γ2, θ2) · φ2 ◦ (γ1, θ1) · φ1 =(γ2s(φ2), φ−12 θ21s(φ2)

)◦(γ1s(φ1), φ−11 θ11s(φ1)

)=

γ2 s(φ2)[t(φ1)−1t(θ1)s(φ1)]︸ ︷︷ ︸t(θ1)s(φ1)

◦γ1s(φ1) , φ−12 θ21s(φ2)φ−11 θ11s(φ1)

(6.19)

(where in the last step we used s(φ2) = t(φ1)) which clearly agrees, in the secondentry, with the right side of (6.18); agreement in the first entry follows usingthe identities:

θ−12 (φ2 ◦ φ1)(4.3)= θ−12 φ11t(φ1)−1 φ2

(4.4)= φ11t(φ1)−1θ−12 φ2. (6.20)

We have thus proved the following.

Theorem 6.1 Let (G,H,α, τ) be a Lie crossed module corresponding to a cate-gorical group K, A a connection form on a principal G-bundle π : P →M . LetPbt(M) and Pbt(P )dec

Abe the categories constructed above, and π : Pbt(P )dec

A→

Pbt(M) the functor given in (6.5). Then π : Pbt(P )decA→ Pbt(M), along with

the right action of K on P defined in (6.6) is a principal categorical bundle.

In section 7 (equation (7.27)) we will construct a categorical principal bundle

π : PdecA→ B

starting from a principal bundle P → B and some additional data. This willgeneralize the specific construction of Theorem 6.1 to provide a ‘decorated’version of a given categorical principal bundle P→ B.

We note that the decoration process is not the same as an associated bundleconstruction (which we consider in section 11), where a quotient is taken of theproduct of a principal bundle and a space on which the structure group acts.

Now recall that from the original principal bundle π : P →M and the con-nection A we also have an undecorated principal categorical bundle Pbt(P )0

Afor which the categorical structure group Gd has object set G and all the mor-phisms are the identity morphisms; the morphisms of Pbt(P )0

Aare simply the

A-horizontal backtrack-erased paths in P . The following result, whose proof isquite clear, is worth noting.

Proposition 6.1 Let G be a categorical group corresponding to a Lie crossedmodule (G,H,α, τ), and Gd be the categorical group whose object set is G andall of whose morphisms are the identity morphisms. Let

Rd : Gd → G

31

be the identity map on objects and the inclusion map on morphisms. Let A be aconnection on a principal G-bundle π : P →M , and let Pbt(P )dec

Aand Pbt(P )0

Abe the principal categorical bundles described above. Consider the association

R : Pbt(P )0A→ Pbt(P )dec

A(6.21)

that, at the level of objects, is the identity map on P and for morphisms is givenby

R(γ) = (eH , γ),

where eH is the identity element in H. Then R is a reduction by Rd, in thesense that R maps each fiber into itself, both on objects and on morphisms, and

R(γφ) = R(γ)Rd(φ)

for all γ ∈ Mor(Pbt(P )0

A

)and φ ∈ Mor(Gd).

7 Categorical connections

By a connection A on a principal categorical bundle π : P→ B, with structurecategorical group K, we mean a prescription for lifting morphisms in B tomorphisms in P. More specifically, for each p ∈ Obj(P) and morphism γ ∈Mor(B), with source π(p), a connection specifies a morphism γhorp : p → q, for

some q with π(q) = t(γ); we call the lift γhorp the lift of γ through p. Of course,we require

π(γhorp

)= γ,

and the lifting should be functorial: if ζ is also a morphism with source π(q)then the lift of ζ ◦ γ through p is

ζhorq ◦ γhorp . (7.1)

Furthermore, we require that a ‘rigid vertical motion’ of a horizontal morphismproduces a horizontal morphism:

γhorp 1g = γhorpg (7.2)

for all g ∈ Obj(G). Identifying morphisms of K with pairs (h, g1) ∈ H oα G,this condition reads

γhorpg = γhorp · (e, g) (7.3)

The morphism γhorp will be called the A-horizontal lift of γ through p. Suchmorphisms will be called A-horizontal morphisms.

In the definition of a connection, as given above, we have imposed a minimalset of rules, to reflect just the situation for traditional bundles and connections.(A general morphism φ ∈ Mor(K) would act on a morphism γhorp : p → q andproduce a morphism pg1 → qg2, where g1 = s(φ) and g2 = t(φ). There is noreason to require that this be horizontal or special in any way.) At first it seems

32

to be too minimal, not involving much role of the morphisms of the structurecategorical group. To bring out the richness and flexibility in the definition wewill develop several examples.

The simplest example arises from an ordinary connection on an ordinaryprincipal bundle π : P → M with structure group G; this is explained inExample CC1 below. To really see the richness of the categorical notion ofconnection, and how the morphism group Mor(G) of a categorical group Gplays a role, we need to pass to decorated bundles obtained from π : P → Mover the path space of M . We summarize the essence of the idea in a compactway here. We start with a principal categorical bundle π : P → B, with astructure categorical group G, associated to a crossed module (G,H,α, τ), andsome additional structure. We construct a decorated bundle, whose objectsare morphisms of P decorated with elements of H. Next we will construct aprincipal categorical bundle P2 → B2 whose structure categorical group G2 hasas objects the morphisms of G:

Obj(G2) = Mor(G). (7.4)

The relation (7.4) means that now we have a principal categorical bundle whosestructure categorical group G2 has as objects the morphisms of the originalstructure categorical group G. The objects of P2 are the morphisms of P1,and so it is Mor(G) that now plays the role that Obj(G) did for P. Thus, aconnection on this new bundle will have behavior controlled by the morphismsof G. Details of this idea are developed through Theorem 7.1, section 10 (forexample, in the context of equation (10.8)). Thus the morphism group of acategorical group can play a significant role in the setting of bundles over pathspaces.

Example CC1. The simplest example is the usual connection on a principalbundle. For this, let A be a connection on a principal bundle π : P → B. LetP0 be the category whose object set is P and for which a morphism p → qis a triple (p, q; γ), where γ is any backtrack-erased path from π(p) to π(q)on B. We take B to be the category with object set B and morphisms beingbacktrack-erased paths. Let G0 have object set G, and have a unique morphismg1 → g2, denoted (g1, g2), for every g1, g2 ∈ G. The action of G0 on P0 is asdescribed in (5.8), and then we have a principal categorical bundle π : P0 → B.For any p ∈ P and any parametrized piecewise C1 path γ on B, correspondingto a morphism of B with source π(p), let γp be the morphism of P0 specifiedby the path γ, the source point p and the target is the point of P obtained byA-parallel-transporting p along γ to its end. Note that this target, and henceγp, is determined by the backtrack-erased form of γ along with p. Thus, wehave a connection A0 on the principal categorical bundle P0 → B.

Example CC2. Let A be a connection form on a principal G-bundle π :P → M , and K a categorical Lie group with Lie crossed module (G,H,α, τ).We now describe a categorical connection on the principal categorical bundleπ : Pbt(P )dec

A→ Pbt(M). Let p ∈ P and γ a parametrized piecewise C1 path on

M with initial point π(p). Define the lift of the morphism γ, with source p, to

33

be(eH , γp),

where γp is the backtrack-erased form of the A-horizontal lift of γ initiating atp, and eH is the identity element in H.

We have the following more general construction of a connection on Pbt(P )decA

.Suppose (G,H,α, τ) is a Lie crossed module corresponding to a categorical Liegroup K, let A be a connection on a principal G-bundle P →M , and let C bean L(H)-valued C∞ 1-form on P that is α-equivariant in the sense that

Cpg(vg) = α(g−1)Cp(v)

for all p ∈ P , g ∈ G, and v ∈ TpP , where pg = Rg(p) is from the right action ofG on P , and vg = R′g(p)v.

Theorem 7.1 We use notation and hypotheses as above. For any backtrack-erased path γ : [t0, t1] → M , and any point u on the fiber over s(γ) = γ(t0),define

γhoru = (γu, hu(γ)) , (7.5)

where γu is the A-horizontal lift of γ through u and hu(γ) is the final point ofthe path [t0, t1]→ H : t 7→ h(t) satisfying the differential equation

h(t)−1h′(t) = −C(γ′u(t)

)(7.6)

at all t ∈ [t0, t1] where γ′(t) exists, with initial point h(t0) = e, the identity inH. Then this lifting process defines a connection on the categorical principalbundle π : Pbt(P )dec

A→ Pbt(M).

The connection ω(A,B) over path space that we have discussed before in(2.14) was determined by an L(G)-valued 1-form A and by τ(B), where B isan L(H)-valued 2-form. We see now that L(H)-valued forms such as C, notintermediated by τ , play a role in connections on decorated bundles.Proof. Replacing u by ug in (7.6), for any g ∈ G, is equal to applying α(g−1),and so, by the uniqueness of a solution of the differential equation, the solutionh(t) gets replaced by α(g−1)h(t) (recall that α(g) is an automorphism of H and(g, h) 7→ α(g−1, h) is smooth). Hence

hug(γ) = α(g−1)hu(γ), (7.7)

and so

(γu, hu(γ)) · (e, g) =(γug, α(g−1)hu(γ)

)= (γug, hug(γ)) , (7.8)

confirming the property (7.3).Now consider backtrack-erased paths γ1 and γ2 on M for which the compos-

ite γ2 ◦ γ1 exists, and let u be in the fiber above s(γ1). Then

γhoru = (γ1,u, h1) ,

34

whereh1 = hu(γ1), (7.9)

has targetv′ = t

(γhor1,u

)= vτ(h1)−1, (7.10)

on using (6.1), where

vdef= t (γ1,u) (7.11)

is the endpoint of γ1,u. The A-horizontal lift of γ2 with initial point v′ is

γ2,v′ = γ2,vτ(h1)−1.

Next note thatγhor2,v = (γ2,v, h2) (7.12)

whereh2 = hv(γ2). (7.13)

Then using (7.10) , (7.8) and the formula for the right action given in (6.6), wehave

γhor2,v′ = γhor2,v · (e, τ(h1)−1)

=(γ2,vτ(h1)−1 , h′2

),

(7.14)

whereh′2 = α(τ(h1))h2. (7.15)

Composing with the lift of γ1 we have

γhor2,v′ ◦ γhor1,u = (γ2,v ◦ γ1,u, h′2h1) . (7.16)

The second term here is clearly

γ2,v ◦ γ1,u = ˜(γ2 ◦ γ1)u.

The first term ish′2h1 = h1h2h

−11 h1 = h1h2.

From the differential equation (7.6) we know that any solution when left mul-tiplied by a constant term in H is again a solution, just with a new initialcondition. Let γ1 have parameter domain [t0, t1] and γ2 have [t1, t2]. Henceh1h2 is the terminal value a(t2) of the solution t 7→ a(t) ∈ H of

a(t)−1a′(t) = −C(γ′2,v(t)

)for t ∈ [t1, t2],

with initial value h1. But h1 itself is the terminal value of the solution of

a(t)−1a′(t) = −C(γ′1,u(t)

)for t ∈ [t0, t1],

35

with initial value a(t0) = e ∈ H. Thus, fitting these two differential equationsinto one, we see that h1h2 is the terminal value a(t2) of the solution a(·) of theequation

a(t)−1a′(t) = −C(γ′u(t)

)for t ∈ [t0, t2],

where γ = γ2 ◦ γ1. Here γ, its A-horizontal lift γ, and a(·) are piecewise C1.Hence

h′2h1 = h1h2 = a(t2) = hu(γ2 ◦ γ1). (7.17)

Recalling the definitions of h1 and h2 from (7.9) and (7.13), let us write thisexplicitly (for future reference and use) as

hu(γ2 ◦ γ1) = hu(γ1)hv(γ2). (7.18)

(We will prove a general version of this observation in Proposition 8.1.) Return-ing to (7.16) we conclude that

γhor2,v′ ◦ γhor1,u = (γu, hu(γ)) = γhoru , (7.19)

where again γ = γ2 ◦ γ1.

Thus γ 7→ γhoru takes composites to composites. QED

A special case of the preceding construction is obtained by taking C of thetype

(dΦ)Φ−1,

for some smooth α-equivariant function Φ : P → H. In this case the horizontallift is given by

hhoru (γ) =(γu,Φ(v)Φ(u)−1

), (7.20)

with usual notation, writing v for the endpoint of γu.Let A0 be a connection on a principal categorical bundle π : P → B with

structure categorical group G0. We will now describe an abstract form of theconstruction used for Pbt(P )dec

Ain Theorem 7.1.

Consider the category PA0whose object set is Obj(P) and morphisms are

A0-horizontal morphisms. The discrete categorical group G0d, whose objectgroup is G = Obj(G0), acts on the right on PA0 by restriction of the originalaction of G0 on P as stated in (7.2). Thus, restricting the projection functor πin the obvious way, we see that

π : PA0→ B (7.21)

is a principal categorical bundle with structure group G0d. Now consider acategorical Lie group G1 whose object group is G. We will construct a principalcategorical bundle with structure categorical group G1 by suitably decoratingthe morphisms of PA0

. Let (G,K,α, τ) be the crossed module correspondingto G1. For the decorated version of PA0 we take as objects just the objects ofP, and as morphisms

(γ, h) ∈ Mor(PA0)×K,

36

where K = ker s1, with s1 being the source map on the morphisms of G2. Wedefine source and targets by

s(γ, h) = s(γ) and t(γ, h) = t(γ)t1(h)−1, (7.22)

and composition by

(γ2, h2) ◦ (γ1, h1) =(γ21τ(h1) ◦ γ1, h2h1

). (7.23)

LetPdec

A0(7.24)

be the category thus defined. Next we define an action of G1 on PdecA0

by

(γ, h) · (h1, g1) =(γ1g1 , α(g−11 )(h−11 h)

), (7.25)

for all (h1, g1) ∈ K ×G, or, equivalently,

(γ, h) · φ =(γ1s(φ), φ

−1h1s(φ)). (7.26)

Thenπ : Pdec

A0→ B (7.27)

is a principal categorical bundle with structure group G1. The proof is the sameas for Theorem 6.1. Thus we have constructed a ‘decorated’ version of a givenprincipal categorical bundle with a given categorical connection.

Let γu denote, as usual, the A0-horizontal lift of γ ∈ Mor(B) with source u.Now suppose k∗ is a map from Mor(PA1

) to K satisfying:

k∗(γu1g1) = α(g−11 )k∗(γu)

k∗(δv ◦ γu

)= k∗ (γu) k∗

(δv

) (7.28)

for all A0-horizontal lift γu, δv for which γv ◦ δu is defined and for all g1 ∈ G.(The notation k∗ is not meant to suggest any type of ‘pullback’.) The firstequality corresponds to (7.7). The second equality above corresponds to theequality (7.18) noted earlier.

Define the horizontal lift of γ ∈ B with initial point u on the fiber over s(γ)to be

hhoru (γ) = (γu, k∗(γu)) , (7.29)

where γu is the A0-horizontal lift of γ with source u.We summarize this construction in the following.

Theorem 7.2 Let A0 be a connection on a principal categorical bundle π :P→ B with structure categorical group G0, and let

PA0 → B

be the corresponding categorical bundle where the morphisms of PA0are the A0-

horizontal morphisms of P. Let (G,K,α, τ) be the crossed module corresponding

37

to a categorical group G1, where the object group of G1 is the same as the objectgroup G of G0. Then the construction described for (7.27) produces a categoricalprincipal bundle

π : PdecA0→ B, (7.30)

with structure group G1.Let k∗ : Mor(PA0

)→ K satisfy the conditions (7.28). Then hhor : Mor(B)→Mor(PA0

) defined by (7.29) specifies a categorical connection A1 on the cate-gorical principal bundle (7.30).

We omit the proof, which is a straightforward abstract reformulation of the firstpart of the proof of Theorem 7.1.

8 Path categories

In this section we set up some definitions and conventions for categories whosemorphisms are paths in some spaces. For the sake of avoiding technicalities thatobscure the main ideas we will focus only on paths that are C∞ and constantnear the endpoints. The latter condition makes it possible to compose two pathswithout losing smoothness at the interface of composition between the paths.

Our first focus is on a generalization of path spaces. Let X be a smoothmanifold. Consider a ‘box’

I =

N∏k=1

[ak, bk] ⊂ RN ,

where N is some positive integer, and ak, bk ∈ R with ak < bk. We denote by

C∞c (I;X)

the set of all C∞ maps I → X with the property that there is some ε > 0such that for any ui ∈ [ai, bi] for i ∈ {1, . . . , N} − {k}, the function uk 7→f(u1, . . . , uN ) is constant when |uk − ak| < ε and when |uk − bk| < ε.

Our interest in the following discussion is in the cases N = 1 and N = 2.Let I0 be the ‘lower face’ of I, and I1 the ‘upper face’ :

I0 =

(N−1∏k=1

[ak, bk]

)× {aN} and I1 =

(N−1∏k=1

[ak, bk]

)× {bN}. (8.1)

Note that if f ∈ C∞c (I;X) then the restrictions f |I0 and f |I1 specify ele-

ments in C∞c (∏N−1k=1 [ak, bk];X), and we think of these elements as the ‘initial’

(source) and ‘final’ (target) values of f . Thus, we think of f ∈ C∞c (I;X) as a‘morphism’ from its source s(f) to its target t(f):

s(f)(x) = f(x, aN ) and t(f)(x) = f(x, bN ). (8.2)

38

Now consider a second box J ⊂ RN , whose lower face J0 is the upper faceI1 of I:

J0 = I1.

In particular, the largest N -th coordinate for points in I is equal to the smallestN -th coordinate for J and I∪J is a box in RN . Then we compose f ∈ C∞c (I;X)with g ∈ C∞c (J ;X) if t(f) = s(g), defining the composite g◦vf to be the elementof C∞c (I ∪ J ;X) given by

g ◦v f(u) =

{f(u) if u ∈ I;

g(u) if v ∈ J .(8.3)

Clearly, the composition operation is associative.Since one should be able to compose a morphism with another if the target

of the first is the source of the second, regardless of the exact domains of themorphisms when taken as maps,

we identify f ∈ C∞c (I;X) and h ∈ C∞c (K;X) if there is some d ∈RN such that I = K + d and f(u) = h(u+ d) for all u ∈ K.

We denote the resulting quotient set by

MapN (X) (8.4)

if the domains of the functions are boxes in RN . Then there are well-definedsource and target maps

s, t : MapN (X)→ MapN−1(X) (8.5)

for all positive integers N . Clearly the composition g ◦v f is meaningful forf, g ∈ MapN (X) if t(f) = s(g). In order to get a category we must haveidentity morphisms and so we quotient one more time, by requiring that

f ◦v i = f and i ◦v g = g (8.6)

whenever the compositions f ◦v i and i ◦v g are meaningful and i ∈ C∞c (I;X) isindependent of the N -th coordinate direction. Let

MorN (X) (8.7)

be the set of all equivalence classes of f ∈ MapN (X), with the equivalence beingdefined by the requirements (8.6). Then the source and target maps s and t in(8.5) descend to well-defined maps

sv, tv : MorN (X)→ MorN−1(X). (8.8)

Again, composition g◦vf is meaningful whenever t(f) = s(g), and the operationof composition is associative. For any a ∈ MorN−1(X) (here N ≥ 1) let 1a ∈MorN (X) be the element with a as both source and target, and 1a, viewed as

39

a mapping on an N -box, is constant along the N -th coordinate direction (thuscorresponding to the i in (8.6)). Then f ◦v 1a = f and g = 1a ◦v g whenevers(f) = a and t(g) = a.

Thus, in this way we obtain, for every positive integer N , a category

PN (X) (8.9)

whose object set is MorN−1(X) and whose morphism set is MorN (X).Our main interest is in the notion of parallel-transport. Parallel-transport

has certain invariance properties: for example, parallel-transport is invariantunder reparametrization of paths and backtrack-erasure. One way of express-ing these properties is to say that parallel-transport is well-defined on cat-egories whose morphisms are obtained by identification of certain classes ofmorphisms in PN (X), such as those that are ‘thin homotopy’ equivalent (asin Theorem 3.2). However, instead of passing to such quotient categories wehave and will state the relevant invariance properties as they arise. The onlyquotients/identification we work with are the bare minimum necessary ones toensure that a category is formed by the maps of interest.

In the following we use terminology introduced in the context of (8.1). Let

X be a manifold, N ≥ 2 a positive integer, I =∏Nj=1[aj , bj ] a box in RN , and

f : I → X a C∞ map. Let C be a C∞ (N − 1)-form on X with values in theLie algebra L(H) of a Lie group H. Consider the solution

wf : [aN , bN ]→ H : u 7→ w(u)

to the differential equation

wf (u)−1w′f (u) = −∫∏N−1

j=1 [aj ,bj ]

C(∂1f(t, u), . . . , ∂N−1f(t, u)

)dt, (8.10)

with initial conditionw(aN ) = e.

We definewC(f) = wf (bN ). (8.11)

With this notation we have the following result on composites.

Proposition 8.1 Let X be a manifold, N ≥ 2 a positive integer, I and J boxesin RN such that the lower face of J is the upper face of I. Let f ∈ C∞c (I;X)and g ∈ C∞c (J ;X) be such that the composite g ◦v f , given by (8.6), is defined.Let C be a C∞ (N − 1)-form on X with values in the Lie algebra L(H) of a Liegroup H. Then

wC(g ◦v f) = wC(f)wC(g). (8.12)

The property (8.12) makes it possible to construct examples of categoricalconnections over path spaces; we have used it in (7.18) and will make use of itagain in a more complex setting in section 9.

40

Proof. Let I be the box∏Nj=1[aj , bj ] and J the box

∏Nj=1[cj , dj ]; since the

upper face of I is the lower face of J , we have bN = cN and [ai, bi] = [ci, di] fori ∈ {1, . . . , N − 1}. The function of u on the right in (8.10) is C∞ and so thedifferential equation (8.10) has a C∞ solution that is uniquely determined bythe initial value wf (aN ). Furthermore, for any h ∈ H the left-translate hwf isthe solution of (8.10) with initial value h. Hence wf (bN )wg is the solution ofthe differential equation

w(u)−1w′(u) = −∫∏N−1

j=1 [cj ,dj ]

C(∂1g(t, u), . . . , ∂N−1g(t, u)

)dt,

with initial condition w(cN ) = wf (bN ). Thus the composite

(wf (bN )wg

)◦ wf : [aN , dN ]→ H : u 7→

{wf (u) if u ∈ [aN , bN ];

wf (bN )wg(u) if u ∈ [cN , dN ]

(8.13)is continuous (with value wf (bN ) at u = bN ), has initial value wf (aN ) = e andis a solution of the differential equation

w(u)−1w′(u) = −∫∏N−1

i=1 [ai,bi]

C(∂1(g ◦v f)(t, u), . . . , ∂N−1(g ◦v f)(t, u)

)dt,

(8.14)at all u ∈ [aN , dN ] except possibly at u = bN . Thus wf (bN )wg agrees with thesolution wg◦vf of (8.14), with initial value e, at all points u ∈ [aN , dN ] exceptpossibly at bN ; continuity of both wf (bN )wg and wg◦vf ensures then that theseare equal also at u = bN . Thus

wg◦vf = wf (bN )wg(u) for all u ∈ [aN , dN ].

Taking u = dN gives

wg◦vf (dN ) = wf (bN )wg(dN ),

which is just the identity (8.12). QED

We can modify wC to another example of a function with a property similarto (8.12):

Proposition 8.2 Let X be a manifold, N ≥ 2 a positive integer. Let C be aC∞ (N − 1)-form on X with values in the Lie algebra L(H) of a Lie group H,and w0 any H-valued function on MorN−1(X):

w0 : MorN−1(X)→ H. (8.15)

Define

wC,0(f) = w0

(s(f)

)wC(f)w0

(t(f)

)−1(8.16)

for all f ∈ C∞c (I;X), where I is any box in RN . Then

wC,0(g ◦v f) = wC,0(f)wC,0(g). (8.17)

for all f ∈ C∞c (I;X) and g ∈ C∞c (J ;X) for which g ◦v f is defined, with I andJ being boxes in RN .

41

Note that in particular we could take for w0 the function wD obtained froman (N − 2)-form D on X with values in H.Proof. The composite g ◦v f has source s(f) and target t(g); hence

wC,0(g ◦v f) = w0

(s(g ◦v f)

)wC(g ◦v f)w0

(t(g ◦v f)

)−1= w0

(s(f)

)wC(f)wC(g)w0

(t(g)

)−1= w0

(s(f)

)wC(f)wD

(t(f)

)−1w0

(t(f)

)wC(g)w0

(t(g)

)−1(using t(f) = s(g).)

= w0

(s(f)

)wC(f)w0

(t(f)

)−1w0

(s(g)

)wC(g)w0

(t(g)

)−1= wC,0(f)wC,0(g).

(8.18)

Thus we have determined the behavior of wC,0 with respect to vertical compo-

sition. QED

9 Categorical connections over path space

We will construct a categorical connection over a path space using the 1-formω(A,B) given in (2.13). For this purpose it will be more convenient to work withC∞ paths that are constant near their endpoints, and define paths of paths as insection 8. As before we work with a Lie crossed module (G,H,α, τ), connectionforms A and A on a principal G-bundle π : P → M , and an L(H)-valued α-equivariant 2-form B on P with values in L(H) that vanishes on (v, w) wheneverv or w is a vertical vector in P .

Let P1(X) and P2(X) be the categories described in section 8. We will workwith the case where X is either M or P .

In Theorem 7.1 we constructed a categorical connection on a categoricalbundle whose objects are points and whose morphisms are paths. We nowconstruct an example that is an analog of this, but one dimension higher in thesense that the objects are now paths and the morphisms are paths of paths. We

focus on the subcategory PA1 (P ) of P1(P ) in which the morphisms arise fromA-horizontal paths on P , and the subcategory P

ω(A,B)

2 (P ) of P2(P ) where themorphisms come from ω(A,B)-horizontal paths of A-horizontal paths on P . Wehave then the categorical principal bundle

π : Pω(A,B)

2 (P )→ P2(M), (9.1)

whose structure categorical group is the categorical group Gd (object set is Gand morphisms are all the identity morphisms). This is the analog of ExampleP3 in section 5, one dimension higher, with paths replaced by paths of paths.The action of any object g ∈ G on any object γ of P

ω(A,B)

2 (P ) produces γg,which is again an A-horizontal path on P . The action of 1g : g → g on Γ ∈Mor

(Pω(A,B)

2 (P ))

produces Γg:

Γ1gdef= Γg,

42

which arises from the map [s0, s1]× [t0, t1]→ P : (s, t) 7→ Γ(s, t)g, where we usea representative map Γ. Note that if Γs = Γs0 for all s ∈ [s0, s1] then Γsg = Γs0gfor all s ∈ [s0, s1].

In order to produce a categorical connection on the bundle (9.1) we haveto provide the rule for horizontal lifts of morphisms in P2(M) to morphismsin P

ω(A,B)

2 (P ). A morphism in P2(M) arises from some Γ : [s0, s1] × [t0, t1] →M : (s, t) 7→ Γs(t); let Γh : [s0, s1] × [t0, t1] → P : (s, t) 7→ Γh

s(t) be its ω(A,B)-

horizontal lift, with a given choice of initial path Γhs0 . It is readily checked that

there is an ε > 0 such that each path Γhs : [t0, t1]→ P is constant within distance

ε from t0 and from t1 (because the same is true for the projected path Γs ofwhich Γh

s is an A-horizontal lift). Moreover, Γhs = Γh

s0 for s near s0, and Γhs = Γh

s1

for s near s1. Furthermore, of course, Γh is C∞. Hence Γh ∈ Mor(Pω(A,B)

2 (P )).

We label this with the given initial path γ = Γhs0 :

Γhγ .

It is clear that the assignmentΓ 7→ Γh

γ

is functorial in the sense that composites of paths are lifted to composites ofthe lifted paths; this is the analog of (7.1), and says that the horizontal lift of∆ ◦v Γ with initial path γ is(

∆ ◦v Γ)hγ

= ∆hδ◦v Γh

γ , (9.2)

whereδ = s

(∆hδ

)= t(

Γhγ

).

(see (8.3) for the definition of ‘vertical’ composition ◦v). The condition of ‘rigidmotion’ (7.2) is also clearly valid since s 7→ Γsg is ω(A,B)-horizontal if s 7→ Γsis ω(A,B)-horizontal (by (2.18)).

10 Parallel-transport for decorated paths

We turn now to our final example of a categorical connection. This will providea connection on a decorated bundle over space of paths. Before turning to thetechnical details let us summarize the essence of the construction. As input wehave a connection A on a principal G-bundle

π : P →M.

Next let G1 be a categorical group with associated crossed module

(G,H,α1, τ1).

Using a connection A and an equivariant 2-form B on P with values in L(H),we have the 1-form

ω(A,B)

43

as specified earlier in (2.13). Now consider a categorical group G2 with

Obj(G2) = Mor(G1),

with associated crossed module

(H oα1G,K,α2, τ2).

We will use this to construct a doubly-decorated category Pdec2 (P )decω(A,B)

whose

objects are of the form(γ, h),

where γ is any A-horizontal path on P and h ∈ H, and whose morphisms areof the form

(Γ, h, k),

where Γ is any ω(A,B)-horizontal path of paths on P . We will show that thereis a principal categorical bundle

Pdec2 (P )decω(A,B)

→ P2(M),

with structure categorical group G2, and then construct categorical connectionson this bundle. Briefly put, our method provides a way of constructing parallel-transport of decorated paths

(γ, h)

along doubly decorated paths of paths

(Γ, h, k),

where γ is any A-horizontal path on the original bundle and Γ is an ω(A,B)-

horizontal path on the bundle of A-horizontal paths over the path space of M .Let us recall the construction provided by Theorem 7.2. Consider a cate-

gorical connection A0 on a categorical principal bundle P→ B, with structurecategorical group G0. Let PA0

→ B be the categorical bundle obtained byworking only with A0-horizontal morphisms of P. Next consider a map

k∗ : Mor(PA0

)→ K,

where K is a group, satisfying

k∗(γu1g0) = α(g0−1)k∗(γu)

k∗(δv ◦ γu

)= k∗ (γu) k∗

(δv

) (10.1)

for all g0 ∈ Obj(G0) and all δv, γu ∈ Mor(PA0

)that are composable. Now let

G1

44

be a categorical group whose object group is the same as the object group ofG0:

Obj(G1) = Obj(G0) = G. (10.2)

Let (G,H,α1, τ1) be the crossed module associated to G1. Theorem 7.2 thenprovides a ‘decorated’ categorical principal bundle Pdec

A0→ B, with structure

group G1 (whose objects are the objects of G0) along with a categorical con-nection A1 on this bundle. Thus, the objects of Pdec

A0are the objects γ of PA0

,but the morphisms are decorated morphisms of PA0

:

(γ, h) ∈ Mor(PA0)×H.

We now apply this procedure with P→ B being a categorical bundle

Pdec2 (P )→ P2(M),

which we now describe.Let π : P →M a principal G-bundle equipped with connection A. Then we

have a categorical principal bundle

PA1 (P )→ P1(M),

where the object set of PA1 (P ) is P and the morphisms arise from A-horizontalpaths γ on P . The structure categorical group G0 is discrete, with object groupbeing G. There is a categorical connection A0 on this bundle: it associates toany γ ∈ Mor

(P1(M)

)the A-horizontal lift γu through any given initial point

u ∈ P in the fiber over s(γ).Now let G1 be a categorical Lie group with crossed module (G,H,α1, τ1).

Then by Theorem 7.2 we have the decorated construction, yielding a categoricalprincipal bundle

PA1 (P )dec → P1(M),

whose object set is P and whose morphisms are of the form

(γ, h),

where γ arises from an A-horizontal path on P , and h ∈ H. The structurecategorical group is G1. We define the category Pdec

2 (P )ω(A,B)as follows. Its

objects are the morphisms (γ, h) of PA1 (P )dec. A morphism of Pdec2 (P )ω(A,B)

isof the form

(Γ, h) ∈ Mor(Pω(A,B)

2 (P ))×H, (10.3)

where Γ is ω(A,B)-horizontal, with source and target being

s(Γ, h) =(s(Γ), h

), and t(Γ, h) =

(t(Γ), h

), (10.4)

where Pω(A,B)

2 (P ) is as we discussed in (9.1). Composition in Pdec2 (P )ω(A,B)

isspecified by

(Γ2, h) ◦ (Γ1, h) = (Γ2 ◦v Γ1, h). (10.5)

45

(We will develop a ‘twisted’ decorated version of this below in (10.12).) Then

Pdec2 (P )ω(A,B)

→ P2(M)

is a categorical principal bundle with structure categorical group G0 being dis-crete, with object group H oα1

G:

Obj(G0) = Mor(G1) = H oα1G.

(We use the notation G0 again in order to make the application of Theorem7.2 clearer.) The action of an object (h1, g1) ∈ Obj(G0) on an object (γ, h) ofPdec

2 (P )ω(A,B)is given by

(γ, h) · (h1, g1) =(γg1, α(g−11 )(h−11 h)

). (10.6)

The action of the identity morphism 1(h1,g1) ∈ Mor(G0) on (Γ, h) is givenby

(Γ, h)1(h1,g1) =(

Γg1, α(g−11 )(h−11 h)). (10.7)

For the categorical connection A0 we take the lifting of Γ through (γ, h) to be

(Γ, h)

where Γ is ω(A,B)-horizontal and has initial path s(Γ) = γ.Now let G2 be a categorical group for which

Obj(G2) = Obj(G0) = Mor(G1) = H oα1 G, (10.8)

with Lie crossed module(H oα1 G,K,α2, τ2). (10.9)

Then by Theorem 7.2 we obtain a doubly decorated categorical bundle

Pdec2 (P )decω(A,B)

→ P2(M), (10.10)

with structure categorical group G2, whose object group is Hoα1G = Obj(G0)

and whose morphism group is K oα2(H oα1

G).A morphism of Pdec

2 (P )decω(A,B)is of the form

(Γ, h, k),

where Γ is ω(A,B)-horizontal, h ∈ H and k ∈ K; its source and target are(obtained from (6.2)):

s(Γ, h, k) =(s(Γ), h) and t(Γ, h, k) =

(t(Γ), h

)τ2(k−1), (10.11)

where in the second term on the right note that τ2(k−1) ∈ H ×α1G acts on the

right on Mor(Pdec1 (P )).

46

The composition of morphisms in this decorated bundle is given (again from(6.2)) by

(∆, h, k) ◦v(Γ, h′, k′) =

((∆, h)τ2(k′) ◦v (Γ, h′), kk′

)(10.12)

(This is the ‘twisted’ decorated version of the composition law (10.5).)The right action of G2 on Pdec

2 (P )decω(A,B)is given as follows. On objects the

action of (h1, g1) ∈ H oα1 G on (γ, h) is

(γ, h)(h1, g1) =(γg1, α1(g−11 )

(h−11 h

)), (10.13)

just as seen before in (7.25). On morphisms, the right action of (k1, h1, g1) ∈K oα2

(H oα1G) on (Γ, h, k) is given by

(Γ, h, k)(k1, h1, g1) =(

Γg1, α1(g−11 )(h−11 h), α2

((h1g1)−1

)(k−11 k)

)(10.14)

Now we will construct a categorical connection on the categorical principalbundle (10.10). To this end assume that we are given a map from the morphismsof the bundle Pdec

2 (P )ω(A,B)(before the K-decoration) to the group K:

k∗ : Mor(Pdec

2 (P )ω(A,B)

)→ K,

satisfying the conditions (10.1):

k∗(Γ1g1) = α2(g−11 )k∗(Γ)

k∗(∆ ◦v Γ) = k∗(Γ)k∗(∆)(10.15)

for all g1 ∈ G, and all ω(A,B)-horizontal Γ and ∆ for which the composite ∆◦v Γis defined. We have seen in Proposition 8.1 how such k∗ may be constructed byusing ‘doubly path-ordered’ integrals of forms with values in the Lie algebra ofK as well as, in Proposition 8.2, how to obtain additional examples by includinga ‘boundary term’ to such integrals. In more detail, suppose C1 is an L(K)-valued 1-form on P , and C2 an L(K)-valued 2-form on P such that they areboth 0 when contracted on a vertical vector and are α2-equivariant in the sensethat

C1((Rg)∗v1) = α2(g)−1C1(v1)

C2

((Rg)∗v1, (Rg)∗v2

)= α2(g−1)C2(v1, v2)

(10.16)

for all g ∈ G, v1, v2 ∈ TpP with p running over P . For Γ : [s0, s1]× [t0, t1]→ P

we define w∗C2(Γ) to be w(t1), where [t0, t1]→ G : t 7→ w(t) solves

w(t)−1w′(t) = −∫ s1

s0

C2

(∂uΓ(u, t), ∂vΓ(u, t)

)du

with w(t0) = e ∈ K. (See equation (8.10).) By α2-equivariance of C2 we thenhave

k∗2(Γg1) = α2(g−11 )k∗2(Γ) (10.17)

47

for all g1 ∈ G. Moreover, k∗2 satisfies the second property of k∗ in (10.15) byProposition 8.1. Now define w1(γ), for any path γ : [s0, s1]→ P , to be w1(s1),where w1 solves

w1(u)−1w′1(u) = −C1

(γ′(u)

),

with w1(s0) = e ∈ K. Then by equivariance of C1 we have

w1(γg1) = α2(g−11 )w1(γ)

for all g1 ∈ G. Finally, set

k∗(Γ) = w1

(s(Γ)

)wC2

(Γ)w1

(t(Γ)

)−1. (10.18)

Then k∗ clearly satisfies the first condition in (10.15) because both w1 and wC2

satisfy this condition; moreover, k∗ also satisfies the second condition in (10.15)by Proposition 8.2.

To construct a connection on the decorated bundle we need to obtain amapping κ∗ similar to k∗ but for the decorated bundle:

κ∗ : Mor(Pdec

2 (P )decω(A,B)

)→ K,

satisfying the conditions (10.1):

κ∗((Γ, h)1(h1,g1)

)= α2((h1g1)−1)κ∗

(Γ, h

)κ∗((∆, h) ◦v

(Γ, h)) = κ∗

(Γ, h

)κ∗(∆, h

),

(10.19)

where, in the second relation, note that for the composition on the left side toexist, the h-component must be the same for ∆ and Γ.

Lemma 10.1 Suppose

k∗ : Mor(Pdec

2 (P )ω(A,B)

)→ K,

satisfies the conditions (10.15). Then the mapping

κ∗ : Mor(Pdec

2 (P )decω(A,B)

)→ K,

specified byκ∗(Γ, h) = α2(h)

(k∗(Γ)

)(10.20)

satisfies (10.19).

Proof. Using the formula for the right action for decorated bundles given in(6.6), we have

(Γ, h)1(h1,g1) =(Γg1, α1(g−11 )(h−11 h)

). (10.21)

48

Applying κ∗ we have

κ∗((Γ, h)1(h1,g1)

)= κ∗

(Γg1, α1(g−11 )(h−11 h)

)= α2

(α1(g−11 )(h−11 h)

)(k∗(Γg1)

)= α2

(α1(g−11 )(h−11 h)

)(α2(g−11 )k∗(Γ)

)(using (10.15))

= α2(g−11 h−11 h)(k∗(Γ)

)(using Lemma 4.2)

= α2

((h1g1)−1

)(α2(h)

(k∗(Γ)

))= α2((h1g1)−1)κ∗

(Γ, h

),

(10.22)

which establishes the first of the relations (10.19).For the second relation we have

κ∗((∆, h) ◦v

(Γ, h)

)= κ∗

(∆ ◦v Γ, h

)(using (10.5))

= α2(h)(k∗(∆ ◦v Γ)

)(using (10.20))

= α2(h)(k∗(Γ)k∗(∆)

)(using (10.15))

= α2(h)(k∗(Γ)

)α2(h)

(k∗(∆)

)= κ∗

(Γ, h

)κ∗(∆, h

). QED

(10.23)

Combining all of this we obtain a categorical connection A2 on the doublydecorated principal bundle Pdec

2 (P )decω(A,B)with structure group G2 whose objects

are the morphisms of an initially given categorical group G1, whose object groupG in turn is the structure group of the original principal bundle π : P → M .Specifically, given a path of paths Γ on M , and an initial A-horizontal path γon P lying above s(Γ), and an element h ∈ H, the result of parallel-transportof (γ, h) by the connection A2 along Γ is the target

t(Γ, h, k),

where Γ is ω(A,B)-horizontal with s(Γ) = γ, and k = κ∗(Γ, h).Previously we have considered a connection ω(A,B) over path space deter-

mined by an L(G)-valued 1-form A and an L(H)-valued 2-form B. In thatsetting ω(A,B) was determined by τ(B) rather than by B itself. The discussionsin this section show that L(H)-valued forms can play a direct role in connectionson decorated bundles.

11 Associated bundles

In the traditional theory of bundles, one constructs bundles that are associatedto a principal bundle through representations of the structure group. In gaugetheory the associated bundles are used to describe matter fields (these are sec-tions of bundles associated to an underlying principal bundle whose structure

49

group is the gauge group). In this section we briefly explore an analog forcategorical principal bundles.

We define a categorical vector space to be a category V analogously to acategorical group. Both Obj(V) and Mor(V) are vector spaces, over some givenfield F. The addition operation

V ×V→ V

is a functor, as is the scalar multiplication

F×V→ V,

where F is the category with object set F and morphisms just the identity mor-phisms for each a ∈ F. (The notion of categorical vector space as defined hereis very restrictive and it may well be worthwhile to consider other structures.)

If K is a categorical group and V a categorical vector space then a repre-sentation ρ of K on V is a functor

ρ : K×V→ V : (k, v) 7→ ρ(k)v

such that ρ(k) : Obj(V)→ Obj(V), for each k ∈ Obj(K) and ρ(φ) : Mor(V)→Mor(V), for all φ ∈ Mor(K), are linear, and ρ gives a representation of the groupObj(K) on the vector space Obj(V) as well as a representation of Mor(K) onMor(V).

Let π : P → B be a principal categorical bundle with group K, and ρa representation of K on a categorical vector space V. Then we construct atwisted product

P×ρ VAn object of this category is an equivalence class [p, v] of pairs (p, v), with twosuch pairs (p′, v′) and (p, v) considered equivalent if p′ = pρ(k) and v′ = ρ(k−1)vfor some k ∈ Obj(K). A morphism of P ×ρ V is an equivalence class [F, f ] ofpairs (F, f), with (F, f) considered equivalent to (F ′, f ′) if F ′ = Fρ(φ) andf ′ = ρ(φ−1)f for some morphism φ of K. The target for [F, f ] is [t(F ), t(f)],and source is [s(F ), s(f)]. There is a well-defined projection functor

P×ρ V→ P

taking [p, v] to πP (p) and [F, f ] to πP (F ). We view this as the associated vectorbundle for the given structures.

Given a connection on P we can construct parallel transport on P×ρ V asfollows. Consider a morphism f : x → y in B and (p, v) ∈ Obj(P) × Obj(V)such that πP (p) = x. Then we define the parallel transport of [p, v] along f tobe [t(F ), v], where F is the horizontal lift of f through p.

We apply the abstract constructions above to the specific principal categor-ical bundles we have studied before.

Let A0 be the categorical connection from Example CC1, associated with aconnection A on a principal G-bundle π : P → B. A morphism of P0 ×ρ V isan equivalence class of ordered pairs/triples

((γ; p, q); v, w),

50

where γ is a backtrack erased path on M from π(p) to π(q), and v, w ∈ V . Wethink of v as being located at the source of γ and w being located at the targetof γ. The equivalence relation between such triples is given by

((γ; p, q); v, w) ∼((γ; pg, qg); g−1(v, w)

). (11.1)

From a connection A on π : P → B we have a connection A0 on π : P0 → B,and then the corresponding parallel-transport of [p, v] along γ results in [q, v],where q is the target of the A-horizontal lift of A initiating at p. This is exactlyin accordance with the traditional notion of parallel transport for associatedbundles.

12 Concluding Remarks

In this paper we have developed a general theory of categorical connectionson categorical bundles and explored several families of examples, including anew class of examples that we call ‘decorated’ bundles. We have explainedhow traditional connection forms along with 2-forms on a classical principalbundle give rise to categorical connections that describe parallel-transport ofdecorated paths over path spaces. Our work is related to that of Martins andPicken [20, 21] but goes beyond the study of holonomies to the study of parallel-transport and we provide new families of examples of these categorical geometricstructures. The large families of examples we have constructed may be special-ized by imposing specific consistency or flatness conditions and would in thatway connect to the notions of categorical connections that are currently in use inthe literature. More precisely, the categorical connections we have constructedin section 10 using 1-forms and 2-forms on a traditional bundle π : P → M ,involving a hierarchy of categorical groups, could be specialized by imposingsuitable flatness constraints on these forms. Our approach also appears to openup the possibility of building a full hierarchy of decorated bundles and connec-tions over spaces of parametrized submanifolds (viewed as paths in lower orderpath spaces) of M of all dimensions.

Acknowledgments. ANS thanks Professor S. Albeverio for discussions andhis kind hospitality at the University of Bonn in 2012, acknowledges researchsupport from a Mercator Guest Professorship at the University of Bonn fromthe Deutsche Forschungsgemeinschaft, and research support from NSA grantH98230-13-1-0210. We are grateful to the referee for perceptive comments thatencouraged us to make alterations that helped improve the presentation.

References

[1] Hossein Abbaspour, Friedrich Wagemann, On 2-Holonomy, (2012) http:

//arxiv.org/abs/1202.2292

[2] Steve Awody, Category Theory, Clarendon Press, Oxford (2006).

51

[3] John C. Baez and James Dolan, Categorification, Contemporary Mathe-matics (Ed. Ezra Getzler and Mikhail Kapranov) Volume 230, AmericanMathematical Society (1998).

[4] John C. Baez and Aaron D. Lauda, Higher-dimensional Algebra V: 2-groups. Theory and Applications of Categories, Vol. 12, No. 14, 2004, pp.423-491.

[5] John C. Baez and Urs Schreiber, Higher Gauge Theory II: 2-Connections,http://math.ucr.edu/home/baez/2conn.pdf

[6] John C. Baez and Urs Schreiber, Higher Gauge Theory: 2-Connections on2-Bundles, http://arxiv.org/abs/hep-th/0412325

[7] John C. Baez and Derek K. Wise, Teleparallel Gravity as a Higher GaugeTheory, http://arxiv.org/abs/1204.4339

[8] John W. Barrett, Holonomy and Path Structures in General Relativity andYang-Mills Theory, International Journal of Theoretical Physics 30 (1991)1171-1215.

[9] John W. Barrett and Marco Mackaay, Categorical Representations of Cat-egorical Groups, Theory and Applications of Categories, Vol. 16 (2006) No.20, 529-557.

[10] Toby Bartels, Higher Gauge Theory I: 2-Bundles, at http://arxiv.org/

abs/math/0410328

[11] Antonio Caetano, and Roger F. Picken, An axiomatic definition of holon-omy, Internat. J. Math. 5 (1994), no. 6, 835848.

[12] Saikat Chatterjee, Amitabha Lahiri, and Ambar N. Sengupta, ParallelTransport over Path Spaces, Reviews in Math. Phys. 9 (2010) 1033-1059.

[13] Magnus Forrester-Barker, Group Objects and Internal Categories, atarXiv:math/0212065v1

[14] Daniel S. Freed, Higher Algebraic Structures and Quantization, Commun.Math. Phys. 159 (1994) 343-398.

[15] K. H. Kamps, D. Pumplun, and W. Tholen (eds.), Category Theory (Gum-mersbach, 1981), no. 962 in Lecture Notes in Mathematics, Berlin, 1982,Springer-Verlag.

[16] Gregory M. Kelly, Basic Concepts of Enriched Categories, Cambridge Uni-versity Press (1982).

[17] Grefory M. Kelly, Ross Street, Reviews of the elements of 2-categories,Springer Lecture Notes in Mathematics, 420, Berlin, (1974), 75-103.

52

[18] Thierry Levy, Two-Dimensional Markovian Holonomy Fields, Asterisque329 (2010).

[19] Saunders Mac Lane, Categories for the Working Mathematician. Springer-Verlag (1971).

[20] Joao Faria Martins and Roger Picken, On two-Dimensional Holonomy,Trans. Amer. Math. Soc. 362 (2010), no. 11, 5657-5695.

[21] Joao Faria Martins and Roger Picken, Surface holonomy for non-abelian2-bundles via double groupoids, Advances in Mathematics 226 (2011), no.4, 3309-3366.

[22] Renee Peiffer, Uber Identitaten zwischen Relationen. Mathematische An-nalen, Vol. 121 (1), 67-99 (1949).

[23] Timothy Porter, Internal categories and crossed modules, in [15], pp.249255.

[24] Urs Schreiber and Konrad Waldorf, Parallel Transport and Functors. J.Homotopy Relat. Struct. 4, 187-244 (2009).

[25] Urs Schreiber and Konrad Waldorf, Connections on non-abelian Gerbesand their Holonomy. http://arxiv.org/pdf/0808.1923v1.pdf

[26] David Viennot, Non-abelian higher gauge theory and categorical bundle,(2012) http://arxiv.org/abs/1202.2280

[27] J. H. C. Whitehead, Combinatorial Homotopy II. Bull. Amer. Math. Soc.Volume 55, Number 5 (1949), 453-496.

[28] Christoph Wockel, Principal 2-bundles and their gauge 2-groups, ForumMath. 23 (2011) no. 3, 565–610.

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